GCSE Maths: Solving quadratic equations by factorising (Higher)

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x + 2)(x + 5) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 2)(x + 5) = 0
   So (x + 2) = 0 or (x + 5) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 2 = 0
   Subtract 2 from both sides: x = −2
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = −2

Check. When x = −5: ((−5) + 2)((−5) + 5) = (−3) × 0 = 0. ✓
When x = −2: ((−2) + 2)((−2) + 5) = 0 × 3 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x − 3)(x + 7) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 3)(x + 7) = 0
   So (x − 3) = 0 or (x + 7) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 3 = 0
   Add 3 to both sides: x = 3
   
   x + 7 = 0
   Subtract 7 from both sides: x = −7
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −7 or x = 3

Check. When x = −7: ((−7) − 3)((−7) + 7) = (−10) × 0 = 0. ✓
When x = 3: (3 − 3)(3 + 7) = 0 × 10 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x − 4)(x − 1) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 4)(x − 1) = 0
   So (x − 4) = 0 or (x − 1) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 4 = 0
   Add 4 to both sides: x = 4
   
   x − 1 = 0
   Add 1 to both sides: x = 1
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 1 or x = 4

Check. When x = 1: (1 − 4)(1 − 1) = (−3) × 0 = 0. ✓
When x = 4: (4 − 4)(4 − 1) = 0 × 3 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   x(x + 6) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x + 6) = 0
   So x = 0 or (x + 6) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   x + 6 = 0
   Subtract 6 from both sides: x = −6
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −6 or x = 0

Check. When x = −6: ((−6))((−6) + 6) = (−6) × 0 = 0. ✓
When x = 0: (0)(0 + 6) = 0 × 6 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x + 8)(x − 8) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 8)(x − 8) = 0
   So (x − 8) = 0 or (x + 8) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 8 = 0
   Add 8 to both sides: x = 8
   
   x + 8 = 0
   Subtract 8 from both sides: x = −8
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −8 or x = 8

Check. When x = −8: ((−8) − 8)((−8) + 8) = (−16) × 0 = 0. ✓
When x = 8: (8 − 8)(8 + 8) = 0 × 16 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x − 5)² = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 5)² = 0
   So (x − 5) = 0.
   A bracket that appears twice gives the same answer twice, so there is only one solution from it (a repeated root).
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 5 = 0
   Add 5 to both sides: x = 5
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 5

Check. When x = 5: (5 − 5)² = 0² = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 6x + 5 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 6 and c = 5.
   We need two numbers that multiply to 5 (the last number) and add to 6 (the middle number). List the pairs that multiply to 5:
   1 × 5 = 5,   add: 1 + 5 = 6 ✓ this pair works
   Put the pair into two brackets: (x + 1)(x + 5).
   Check by expanding: x² + 5x + 1x + 5 = x² + 6x + 5. ✓
   x² + 6x + 5 = (x + 1)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 1)(x + 5) = 0
   So (x + 1) = 0 or (x + 5) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 1 = 0
   Subtract 1 from both sides: x = −1
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = −1

Check. When x = −5: left side = (−5)² + 6 × (−5) + 5 = 25 − 30 + 5 = 0. It equals 0. ✓
When x = −1: left side = (−1)² + 6 × (−1) + 5 = 1 − 6 + 5 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 7x + 12 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 7 and c = 12.
   We need two numbers that multiply to 12 (the last number) and add to 7 (the middle number). List the pairs that multiply to 12:
   1 × 12 = 12,   add: 1 + 12 = 13 ✗
   -1 × -12 = 12,   add: -1 + -12 = -13 ✗
   2 × 6 = 12,   add: 2 + 6 = 8 ✗
   -2 × -6 = 12,   add: -2 + -6 = -8 ✗
   3 × 4 = 12,   add: 3 + 4 = 7 ✓ this pair works
   Put the pair into two brackets: (x + 3)(x + 4).
   Check by expanding: x² + 4x + 3x + 12 = x² + 7x + 12. ✓
   x² + 7x + 12 = (x + 3)(x + 4)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 3)(x + 4) = 0
   So (x + 3) = 0 or (x + 4) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 3 = 0
   Subtract 3 from both sides: x = −3
   
   x + 4 = 0
   Subtract 4 from both sides: x = −4
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −4 or x = −3

Check. When x = −4: left side = (−4)² + 7 × (−4) + 12 = 16 − 28 + 12 = 0. It equals 0. ✓
When x = −3: left side = (−3)² + 7 × (−3) + 12 = 9 − 21 + 12 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 8x + 15 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 8 and c = 15.
   We need two numbers that multiply to 15 (the last number) and add to 8 (the middle number). List the pairs that multiply to 15:
   1 × 15 = 15,   add: 1 + 15 = 16 ✗
   -1 × -15 = 15,   add: -1 + -15 = -16 ✗
   3 × 5 = 15,   add: 3 + 5 = 8 ✓ this pair works
   Put the pair into two brackets: (x + 3)(x + 5).
   Check by expanding: x² + 5x + 3x + 15 = x² + 8x + 15. ✓
   x² + 8x + 15 = (x + 3)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 3)(x + 5) = 0
   So (x + 3) = 0 or (x + 5) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 3 = 0
   Subtract 3 from both sides: x = −3
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = −3

Check. When x = −5: left side = (−5)² + 8 × (−5) + 15 = 25 − 40 + 15 = 0. It equals 0. ✓
When x = −3: left side = (−3)² + 8 × (−3) + 15 = 9 − 24 + 15 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 9x + 14 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 9 and c = 14.
   We need two numbers that multiply to 14 (the last number) and add to 9 (the middle number). List the pairs that multiply to 14:
   1 × 14 = 14,   add: 1 + 14 = 15 ✗
   -1 × -14 = 14,   add: -1 + -14 = -15 ✗
   2 × 7 = 14,   add: 2 + 7 = 9 ✓ this pair works
   Put the pair into two brackets: (x + 2)(x + 7).
   Check by expanding: x² + 7x + 2x + 14 = x² + 9x + 14. ✓
   x² + 9x + 14 = (x + 2)(x + 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 2)(x + 7) = 0
   So (x + 2) = 0 or (x + 7) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 2 = 0
   Subtract 2 from both sides: x = −2
   
   x + 7 = 0
   Subtract 7 from both sides: x = −7
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −7 or x = −2

Check. When x = −7: left side = (−7)² + 9 × (−7) + 14 = 49 − 63 + 14 = 0. It equals 0. ✓
When x = −2: left side = (−2)² + 9 × (−2) + 14 = 4 − 18 + 14 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 11x + 10 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 11 and c = 10.
   We need two numbers that multiply to 10 (the last number) and add to 11 (the middle number). List the pairs that multiply to 10:
   1 × 10 = 10,   add: 1 + 10 = 11 ✓ this pair works
   Put the pair into two brackets: (x + 1)(x + 10).
   Check by expanding: x² + 10x + 1x + 10 = x² + 11x + 10. ✓
   x² + 11x + 10 = (x + 1)(x + 10)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 1)(x + 10) = 0
   So (x + 1) = 0 or (x + 10) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 1 = 0
   Subtract 1 from both sides: x = −1
   
   x + 10 = 0
   Subtract 10 from both sides: x = −10
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −10 or x = −1

Check. When x = −10: left side = (−10)² + 11 × (−10) + 10 = 100 − 110 + 10 = 0. It equals 0. ✓
When x = −1: left side = (−1)² + 11 × (−1) + 10 = 1 − 11 + 10 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 10x + 21 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 10 and c = 21.
   We need two numbers that multiply to 21 (the last number) and add to 10 (the middle number). List the pairs that multiply to 21:
   1 × 21 = 21,   add: 1 + 21 = 22 ✗
   -1 × -21 = 21,   add: -1 + -21 = -22 ✗
   3 × 7 = 21,   add: 3 + 7 = 10 ✓ this pair works
   Put the pair into two brackets: (x + 3)(x + 7).
   Check by expanding: x² + 7x + 3x + 21 = x² + 10x + 21. ✓
   x² + 10x + 21 = (x + 3)(x + 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 3)(x + 7) = 0
   So (x + 3) = 0 or (x + 7) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 3 = 0
   Subtract 3 from both sides: x = −3
   
   x + 7 = 0
   Subtract 7 from both sides: x = −7
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −7 or x = −3

Check. When x = −7: left side = (−7)² + 10 × (−7) + 21 = 49 − 70 + 21 = 0. It equals 0. ✓
When x = −3: left side = (−3)² + 10 × (−3) + 21 = 9 − 30 + 21 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 7x + 10 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -7 and c = 10.
   We need two numbers that multiply to 10 (the last number) and add to -7 (the middle number). List the pairs that multiply to 10:
   1 × 10 = 10,   add: 1 + 10 = 11 ✗
   -1 × -10 = 10,   add: -1 + -10 = -11 ✗
   2 × 5 = 10,   add: 2 + 5 = 7 ✗
   -2 × -5 = 10,   add: -2 + -5 = -7 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x − 5).
   Check by expanding: x² − 5x − 2x + 10 = x² − 7x + 10. ✓
   x² − 7x + 10 = (x − 2)(x − 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x − 5) = 0
   So (x − 5) = 0 or (x − 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 5 = 0
   Add 5 to both sides: x = 5
   
   x − 2 = 0
   Add 2 to both sides: x = 2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 2 or x = 5

Check. When x = 2: left side = 2² − 7 × 2 + 10 = 4 − 14 + 10 = 0. It equals 0. ✓
When x = 5: left side = 5² − 7 × 5 + 10 = 25 − 35 + 10 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 6x + 8 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -6 and c = 8.
   We need two numbers that multiply to 8 (the last number) and add to -6 (the middle number). List the pairs that multiply to 8:
   1 × 8 = 8,   add: 1 + 8 = 9 ✗
   -1 × -8 = 8,   add: -1 + -8 = -9 ✗
   2 × 4 = 8,   add: 2 + 4 = 6 ✗
   -2 × -4 = 8,   add: -2 + -4 = -6 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x − 4).
   Check by expanding: x² − 4x − 2x + 8 = x² − 6x + 8. ✓
   x² − 6x + 8 = (x − 2)(x − 4)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x − 4) = 0
   So (x − 4) = 0 or (x − 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 4 = 0
   Add 4 to both sides: x = 4
   
   x − 2 = 0
   Add 2 to both sides: x = 2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 2 or x = 4

Check. When x = 2: left side = 2² − 6 × 2 + 8 = 4 − 12 + 8 = 0. It equals 0. ✓
When x = 4: left side = 4² − 6 × 4 + 8 = 16 − 24 + 8 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 8x + 12 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -8 and c = 12.
   We need two numbers that multiply to 12 (the last number) and add to -8 (the middle number). List the pairs that multiply to 12:
   1 × 12 = 12,   add: 1 + 12 = 13 ✗
   -1 × -12 = 12,   add: -1 + -12 = -13 ✗
   2 × 6 = 12,   add: 2 + 6 = 8 ✗
   -2 × -6 = 12,   add: -2 + -6 = -8 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x − 6).
   Check by expanding: x² − 6x − 2x + 12 = x² − 8x + 12. ✓
   x² − 8x + 12 = (x − 2)(x − 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x − 6) = 0
   So (x − 6) = 0 or (x − 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 6 = 0
   Add 6 to both sides: x = 6
   
   x − 2 = 0
   Add 2 to both sides: x = 2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 2 or x = 6

Check. When x = 2: left side = 2² − 8 × 2 + 12 = 4 − 16 + 12 = 0. It equals 0. ✓
When x = 6: left side = 6² − 8 × 6 + 12 = 36 − 48 + 12 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 9x + 18 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -9 and c = 18.
   We need two numbers that multiply to 18 (the last number) and add to -9 (the middle number). List the pairs that multiply to 18:
   1 × 18 = 18,   add: 1 + 18 = 19 ✗
   -1 × -18 = 18,   add: -1 + -18 = -19 ✗
   2 × 9 = 18,   add: 2 + 9 = 11 ✗
   -2 × -9 = 18,   add: -2 + -9 = -11 ✗
   3 × 6 = 18,   add: 3 + 6 = 9 ✗
   -3 × -6 = 18,   add: -3 + -6 = -9 ✓ this pair works
   Put the pair into two brackets: (x − 3)(x − 6).
   Check by expanding: x² − 6x − 3x + 18 = x² − 9x + 18. ✓
   x² − 9x + 18 = (x − 3)(x − 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 3)(x − 6) = 0
   So (x − 6) = 0 or (x − 3) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 6 = 0
   Add 6 to both sides: x = 6
   
   x − 3 = 0
   Add 3 to both sides: x = 3
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 3 or x = 6

Check. When x = 3: left side = 3² − 9 × 3 + 18 = 9 − 27 + 18 = 0. It equals 0. ✓
When x = 6: left side = 6² − 9 × 6 + 18 = 36 − 54 + 18 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 10x + 16 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -10 and c = 16.
   We need two numbers that multiply to 16 (the last number) and add to -10 (the middle number). List the pairs that multiply to 16:
   1 × 16 = 16,   add: 1 + 16 = 17 ✗
   -1 × -16 = 16,   add: -1 + -16 = -17 ✗
   2 × 8 = 16,   add: 2 + 8 = 10 ✗
   -2 × -8 = 16,   add: -2 + -8 = -10 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x − 8).
   Check by expanding: x² − 8x − 2x + 16 = x² − 10x + 16. ✓
   x² − 10x + 16 = (x − 2)(x − 8)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x − 8) = 0
   So (x − 8) = 0 or (x − 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 8 = 0
   Add 8 to both sides: x = 8
   
   x − 2 = 0
   Add 2 to both sides: x = 2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 2 or x = 8

Check. When x = 2: left side = 2² − 10 × 2 + 16 = 4 − 20 + 16 = 0. It equals 0. ✓
When x = 8: left side = 8² − 10 × 8 + 16 = 64 − 80 + 16 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 11x + 24 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -11 and c = 24.
   We need two numbers that multiply to 24 (the last number) and add to -11 (the middle number). List the pairs that multiply to 24:
   1 × 24 = 24,   add: 1 + 24 = 25 ✗
   -1 × -24 = 24,   add: -1 + -24 = -25 ✗
   2 × 12 = 24,   add: 2 + 12 = 14 ✗
   -2 × -12 = 24,   add: -2 + -12 = -14 ✗
   3 × 8 = 24,   add: 3 + 8 = 11 ✗
   -3 × -8 = 24,   add: -3 + -8 = -11 ✓ this pair works
   Put the pair into two brackets: (x − 3)(x − 8).
   Check by expanding: x² − 8x − 3x + 24 = x² − 11x + 24. ✓
   x² − 11x + 24 = (x − 3)(x − 8)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 3)(x − 8) = 0
   So (x − 8) = 0 or (x − 3) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 8 = 0
   Add 8 to both sides: x = 8
   
   x − 3 = 0
   Add 3 to both sides: x = 3
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 3 or x = 8

Check. When x = 3: left side = 3² − 11 × 3 + 24 = 9 − 33 + 24 = 0. It equals 0. ✓
When x = 8: left side = 8² − 11 × 8 + 24 = 64 − 88 + 24 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 2x − 8 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 2 and c = -8.
   We need two numbers that multiply to -8 (the last number) and add to 2 (the middle number). List the pairs that multiply to -8:
   1 × -8 = -8,   add: 1 + -8 = -7 ✗
   -1 × 8 = -8,   add: -1 + 8 = 7 ✗
   2 × -4 = -8,   add: 2 + -4 = -2 ✗
   -2 × 4 = -8,   add: -2 + 4 = 2 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x + 4).
   Check by expanding: x² + 4x − 2x − 8 = x² + 2x − 8. ✓
   x² + 2x − 8 = (x − 2)(x + 4)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x + 4) = 0
   So (x − 2) = 0 or (x + 4) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 2 = 0
   Add 2 to both sides: x = 2
   
   x + 4 = 0
   Subtract 4 from both sides: x = −4
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −4 or x = 2

Check. When x = −4: left side = (−4)² + 2 × (−4) − 8 = 16 − 8 − 8 = 0. It equals 0. ✓
When x = 2: left side = 2² + 2 × 2 − 8 = 4 + 4 − 8 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − x − 6 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -1 and c = -6.
   We need two numbers that multiply to -6 (the last number) and add to -1 (the middle number). List the pairs that multiply to -6:
   1 × -6 = -6,   add: 1 + -6 = -5 ✗
   -1 × 6 = -6,   add: -1 + 6 = 5 ✗
   2 × -3 = -6,   add: 2 + -3 = -1 ✓ this pair works
   Put the pair into two brackets: (x + 2)(x − 3).
   Check by expanding: x² − 3x + 2x − 6 = x² − x − 6. ✓
   x² − x − 6 = (x + 2)(x − 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 2)(x − 3) = 0
   So (x − 3) = 0 or (x + 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 3 = 0
   Add 3 to both sides: x = 3
   
   x + 2 = 0
   Subtract 2 from both sides: x = −2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −2 or x = 3

Check. When x = −2: left side = (−2)² − (−2) − 6 = 4 + 2 − 6 = 0. It equals 0. ✓
When x = 3: left side = 3² − 3 − 6 = 9 − 3 − 6 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 3x − 10 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 3 and c = -10.
   We need two numbers that multiply to -10 (the last number) and add to 3 (the middle number). List the pairs that multiply to -10:
   1 × -10 = -10,   add: 1 + -10 = -9 ✗
   -1 × 10 = -10,   add: -1 + 10 = 9 ✗
   2 × -5 = -10,   add: 2 + -5 = -3 ✗
   -2 × 5 = -10,   add: -2 + 5 = 3 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x + 5).
   Check by expanding: x² + 5x − 2x − 10 = x² + 3x − 10. ✓
   x² + 3x − 10 = (x − 2)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x + 5) = 0
   So (x − 2) = 0 or (x + 5) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 2 = 0
   Add 2 to both sides: x = 2
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = 2

Check. When x = −5: left side = (−5)² + 3 × (−5) − 10 = 25 − 15 − 10 = 0. It equals 0. ✓
When x = 2: left side = 2² + 3 × 2 − 10 = 4 + 6 − 10 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 4x − 12 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -4 and c = -12.
   We need two numbers that multiply to -12 (the last number) and add to -4 (the middle number). List the pairs that multiply to -12:
   1 × -12 = -12,   add: 1 + -12 = -11 ✗
   -1 × 12 = -12,   add: -1 + 12 = 11 ✗
   2 × -6 = -12,   add: 2 + -6 = -4 ✓ this pair works
   Put the pair into two brackets: (x + 2)(x − 6).
   Check by expanding: x² − 6x + 2x − 12 = x² − 4x − 12. ✓
   x² − 4x − 12 = (x + 2)(x − 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 2)(x − 6) = 0
   So (x − 6) = 0 or (x + 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 6 = 0
   Add 6 to both sides: x = 6
   
   x + 2 = 0
   Subtract 2 from both sides: x = −2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −2 or x = 6

Check. When x = −2: left side = (−2)² − 4 × (−2) − 12 = 4 + 8 − 12 = 0. It equals 0. ✓
When x = 6: left side = 6² − 4 × 6 − 12 = 36 − 24 − 12 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + x − 20 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 1 and c = -20.
   We need two numbers that multiply to -20 (the last number) and add to 1 (the middle number). List the pairs that multiply to -20:
   1 × -20 = -20,   add: 1 + -20 = -19 ✗
   -1 × 20 = -20,   add: -1 + 20 = 19 ✗
   2 × -10 = -20,   add: 2 + -10 = -8 ✗
   -2 × 10 = -20,   add: -2 + 10 = 8 ✗
   4 × -5 = -20,   add: 4 + -5 = -1 ✗
   -4 × 5 = -20,   add: -4 + 5 = 1 ✓ this pair works
   Put the pair into two brackets: (x − 4)(x + 5).
   Check by expanding: x² + 5x − 4x − 20 = x² + x − 20. ✓
   x² + x − 20 = (x − 4)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 4)(x + 5) = 0
   So (x − 4) = 0 or (x + 5) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 4 = 0
   Add 4 to both sides: x = 4
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = 4

Check. When x = −5: left side = (−5)² + (−5) − 20 = 25 − 5 − 20 = 0. It equals 0. ✓
When x = 4: left side = 4² + 4 − 20 = 16 + 4 − 20 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 2x − 24 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -2 and c = -24.
   We need two numbers that multiply to -24 (the last number) and add to -2 (the middle number). List the pairs that multiply to -24:
   1 × -24 = -24,   add: 1 + -24 = -23 ✗
   -1 × 24 = -24,   add: -1 + 24 = 23 ✗
   2 × -12 = -24,   add: 2 + -12 = -10 ✗
   -2 × 12 = -24,   add: -2 + 12 = 10 ✗
   3 × -8 = -24,   add: 3 + -8 = -5 ✗
   -3 × 8 = -24,   add: -3 + 8 = 5 ✗
   4 × -6 = -24,   add: 4 + -6 = -2 ✓ this pair works
   Put the pair into two brackets: (x + 4)(x − 6).
   Check by expanding: x² − 6x + 4x − 24 = x² − 2x − 24. ✓
   x² − 2x − 24 = (x + 4)(x − 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 4)(x − 6) = 0
   So (x − 6) = 0 or (x + 4) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 6 = 0
   Add 6 to both sides: x = 6
   
   x + 4 = 0
   Subtract 4 from both sides: x = −4
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −4 or x = 6

Check. When x = −4: left side = (−4)² − 2 × (−4) − 24 = 16 + 8 − 24 = 0. It equals 0. ✓
When x = 6: left side = 6² − 2 × 6 − 24 = 36 − 12 − 24 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 3x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², 3x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   3x ÷ x = 3
   So x² + 3x = x(x + 3).
   Check by expanding: x × (x + 3) = x × x, x × 3 → x² + 3x. ✓
   x² + 3x = x(x + 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x + 3) = 0
   So x = 0 or (x + 3) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   x + 3 = 0
   Subtract 3 from both sides: x = −3
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −3 or x = 0

Check. When x = −3: left side = (−3)² + 3 × (−3) = 9 − 9 = 0. It equals 0. ✓
When x = 0: left side = 0² + 3 × 0 = 0 + 0 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 7x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², −7x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   −7x ÷ x = −7
   So x² − 7x = x(x − 7).
   Check by expanding: x × (x − 7) = x × x, x × (−7) → x² − 7x. ✓
   x² − 7x = x(x − 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x − 7) = 0
   So (x − 7) = 0 or x = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 7 = 0
   Add 7 to both sides: x = 7
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 7

Check. When x = 0: left side = 0² − 7 × 0 = 0 + 0 = 0. It equals 0. ✓
When x = 7: left side = 7² − 7 × 7 = 49 − 49 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 10x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², 10x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   10x ÷ x = 10
   So x² + 10x = x(x + 10).
   Check by expanding: x × (x + 10) = x × x, x × 10 → x² + 10x. ✓
   x² + 10x = x(x + 10)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x + 10) = 0
   So x = 0 or (x + 10) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   x + 10 = 0
   Subtract 10 from both sides: x = −10
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −10 or x = 0

Check. When x = −10: left side = (−10)² + 10 × (−10) = 100 − 100 = 0. It equals 0. ✓
When x = 0: left side = 0² + 10 × 0 = 0 + 0 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   2x² − 8x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 2x², −8x. The biggest number and the highest power of x that divide into all of them is 2x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   2x² ÷ 2x = x
   −8x ÷ 2x = −4
   So 2x² − 8x = 2x(x − 4).
   Check by expanding: 2x × (x − 4) = 2x × x, 2x × (−4) → 2x² − 8x. ✓
   2x² − 8x = 2x(x − 4)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   2x(x − 4) = 0
   So (x − 4) = 0 or x = 0.
   The number 2 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 4 = 0
   Add 4 to both sides: x = 4
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 4

Check. When x = 0: left side = 2 × 0² − 8 × 0 = 0 + 0 = 0. It equals 0. ✓
When x = 4: left side = 2 × 4² − 8 × 4 = 32 − 32 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   3x² + 9x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 3x², 9x. The biggest number and the highest power of x that divide into all of them is 3x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   3x² ÷ 3x = x
   9x ÷ 3x = 3
   So 3x² + 9x = 3x(x + 3).
   Check by expanding: 3x × (x + 3) = 3x × x, 3x × 3 → 3x² + 9x. ✓
   3x² + 9x = 3x(x + 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   3x(x + 3) = 0
   So x = 0 or (x + 3) = 0.
   The number 3 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   x + 3 = 0
   Subtract 3 from both sides: x = −3
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −3 or x = 0

Check. When x = −3: left side = 3 × (−3)² + 9 × (−3) = 27 − 27 = 0. It equals 0. ✓
When x = 0: left side = 3 × 0² + 9 × 0 = 0 + 0 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², −x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   −x ÷ x = −1
   So x² − x = x(x − 1).
   Check by expanding: x × (x − 1) = x × x, x × (−1) → x² − x. ✓
   x² − x = x(x − 1)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x − 1) = 0
   So (x − 1) = 0 or x = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 1 = 0
   Add 1 to both sides: x = 1
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 1

Check. When x = 0: left side = 0² − 0 = 0 + 0 = 0. It equals 0. ✓
When x = 1: left side = 1² − 1 = 1 − 1 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 4 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 4 = 2², so b = 2.
   Put them into the rule: (x − 2)(x + 2).
   Check by expanding: (x − 2)(x + 2) = x² + 2x − 2x − 4 = x² − 4. ✓ (the middle terms cancel)
   x² − 4 = (x − 2)(x + 2)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x + 2) = 0
   So (x − 2) = 0 or (x + 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 2 = 0
   Add 2 to both sides: x = 2
   
   x + 2 = 0
   Subtract 2 from both sides: x = −2
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −2 or x = 2

Check. When x = −2: left side = (−2)² − 4 = 4 − 4 = 0. It equals 0. ✓
When x = 2: left side = 2² − 4 = 4 − 4 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 25 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 25 = 5², so b = 5.
   Put them into the rule: (x − 5)(x + 5).
   Check by expanding: (x − 5)(x + 5) = x² + 5x − 5x − 25 = x² − 25. ✓ (the middle terms cancel)
   x² − 25 = (x − 5)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 5)(x + 5) = 0
   So (x − 5) = 0 or (x + 5) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 5 = 0
   Add 5 to both sides: x = 5
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = 5

Check. When x = −5: left side = (−5)² − 25 = 25 − 25 = 0. It equals 0. ✓
When x = 5: left side = 5² − 25 = 25 − 25 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 64 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 64 = 8², so b = 8.
   Put them into the rule: (x − 8)(x + 8).
   Check by expanding: (x − 8)(x + 8) = x² + 8x − 8x − 64 = x² − 64. ✓ (the middle terms cancel)
   x² − 64 = (x − 8)(x + 8)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 8)(x + 8) = 0
   So (x − 8) = 0 or (x + 8) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 8 = 0
   Add 8 to both sides: x = 8
   
   x + 8 = 0
   Subtract 8 from both sides: x = −8
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −8 or x = 8

Check. When x = −8: left side = (−8)² − 64 = 64 − 64 = 0. It equals 0. ✓
When x = 8: left side = 8² − 64 = 64 − 64 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 121 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 121 = 11², so b = 11.
   Put them into the rule: (x − 11)(x + 11).
   Check by expanding: (x − 11)(x + 11) = x² + 11x − 11x − 121 = x² − 121. ✓ (the middle terms cancel)
   x² − 121 = (x − 11)(x + 11)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 11)(x + 11) = 0
   So (x − 11) = 0 or (x + 11) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 11 = 0
   Add 11 to both sides: x = 11
   
   x + 11 = 0
   Subtract 11 from both sides: x = −11
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −11 or x = 11

Check. When x = −11: left side = (−11)² − 121 = 121 − 121 = 0. It equals 0. ✓
When x = 11: left side = 11² − 121 = 121 − 121 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 1 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 1 = 1², so b = 1.
   Put them into the rule: (x − 1)(x + 1).
   Check by expanding: (x − 1)(x + 1) = x² + 1x − 1x − 1 = x² − 1. ✓ (the middle terms cancel)
   x² − 1 = (x − 1)(x + 1)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 1)(x + 1) = 0
   So (x − 1) = 0 or (x + 1) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 1 = 0
   Add 1 to both sides: x = 1
   
   x + 1 = 0
   Subtract 1 from both sides: x = −1
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −1 or x = 1

Check. When x = −1: left side = (−1)² − 1 = 1 − 1 = 0. It equals 0. ✓
When x = 1: left side = 1² − 1 = 1 − 1 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 400 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 400 = 20², so b = 20.
   Put them into the rule: (x − 20)(x + 20).
   Check by expanding: (x − 20)(x + 20) = x² + 20x − 20x − 400 = x² − 400. ✓ (the middle terms cancel)
   x² − 400 = (x − 20)(x + 20)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 20)(x + 20) = 0
   So (x − 20) = 0 or (x + 20) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 20 = 0
   Add 20 to both sides: x = 20
   
   x + 20 = 0
   Subtract 20 from both sides: x = −20
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −20 or x = 20

Check. When x = −20: left side = (−20)² − 400 = 400 − 400 = 0. It equals 0. ✓
When x = 20: left side = 20² − 400 = 400 − 400 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Method. Count the solutions: each different bracket that can be 0 gives one solution.

1. Look at the equation. Use the null factor law.
   (x − 1)(x + 4) = 0
2. Count. The brackets (x − 1) and (x + 4) are different, so there are two solutions: x = 1 and x = −4.
   2 solutions

Watch out. Two identical brackets give only one solution.

Method. Count the solutions: each different bracket that can be 0 gives one solution.

1. Look at the equation. Use the null factor law.
   (x − 6)² = 0
2. Count. Both brackets are the same, (x − 6), so they give the same answer, x = 6.
   1 solution

Watch out. Two identical brackets give only one solution.

Method. Count the solutions: each different bracket that can be 0 gives one solution.

1. Look at the equation. Use the null factor law.
   x(x − 2) = 0
2. Count. Either x = 0 or x − 2 = 0, which gives x = 0 and x = 2.
   2 solutions

Watch out. Two identical brackets give only one solution.

Method. Rearrange to make x² the subject, then think about squaring.

1. Subtract 4 from both sides. To get x² on its own.
   x² = −4
2. Think about squares. A positive number times itself is positive, a negative number times itself is positive, and 0 × 0 = 0. So a square number is never negative.
   x² can never equal −4
3. Conclude. No real number satisfies the equation.
   0 solutions

Check. The graph of y = x² + 4 stays above the x-axis (its lowest point is (0, 4)), so it never crosses it. ✓

Watch out. You cannot factorise x² + 4 into two brackets, because it is a sum of two squares.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 9x + 18 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 9 and c = 18.
   We need two numbers that multiply to 18 (the last number) and add to 9 (the middle number). List the pairs that multiply to 18:
   1 × 18 = 18,   add: 1 + 18 = 19 ✗
   -1 × -18 = 18,   add: -1 + -18 = -19 ✗
   2 × 9 = 18,   add: 2 + 9 = 11 ✗
   -2 × -9 = 18,   add: -2 + -9 = -11 ✗
   3 × 6 = 18,   add: 3 + 6 = 9 ✓ this pair works
   Put the pair into two brackets: (x + 3)(x + 6).
   Check by expanding: x² + 6x + 3x + 18 = x² + 9x + 18. ✓
   x² + 9x + 18 = (x + 3)(x + 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x + 3)(x + 6) = 0
   So (x + 3) = 0 or (x + 6) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 3 = 0
   Subtract 3 from both sides: x = −3
   
   x + 6 = 0
   Subtract 6 from both sides: x = −6
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −6 or x = −3

Check. When x = −6: left side = (−6)² + 9 × (−6) + 18 = 36 − 54 + 18 = 0. It equals 0. ✓
When x = −3: left side = (−3)² + 9 × (−3) + 18 = 9 − 27 + 18 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 3x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², −3x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   −3x ÷ x = −3
   So x² − 3x = x(x − 3).
   Check by expanding: x × (x − 3) = x × x, x × (−3) → x² − 3x. ✓
   x² − 3x = x(x − 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x − 3) = 0
   So (x − 3) = 0 or x = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 3 = 0
   Add 3 to both sides: x = 3
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 3

Check. When x = 0: left side = 0² − 3 × 0 = 0 + 0 = 0. It equals 0. ✓
When x = 3: left side = 3² − 3 × 3 = 9 − 9 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 49 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 49 = 7², so b = 7.
   Put them into the rule: (x − 7)(x + 7).
   Check by expanding: (x − 7)(x + 7) = x² + 7x − 7x − 49 = x² − 49. ✓ (the middle terms cancel)
   x² − 49 = (x − 7)(x + 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 7)(x + 7) = 0
   So (x − 7) = 0 or (x + 7) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 7 = 0
   Add 7 to both sides: x = 7
   
   x + 7 = 0
   Subtract 7 from both sides: x = −7
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −7 or x = 7

Check. When x = −7: left side = (−7)² − 49 = 49 − 49 = 0. It equals 0. ✓
When x = 7: left side = 7² − 49 = 49 − 49 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 8x + 7 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -8 and c = 7.
   We need two numbers that multiply to 7 (the last number) and add to -8 (the middle number). List the pairs that multiply to 7:
   1 × 7 = 7,   add: 1 + 7 = 8 ✗
   -1 × -7 = 7,   add: -1 + -7 = -8 ✓ this pair works
   Put the pair into two brackets: (x − 1)(x − 7).
   Check by expanding: x² − 7x − 1x + 7 = x² − 8x + 7. ✓
   x² − 8x + 7 = (x − 1)(x − 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 1)(x − 7) = 0
   So (x − 7) = 0 or (x − 1) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 7 = 0
   Add 7 to both sides: x = 7
   
   x − 1 = 0
   Add 1 to both sides: x = 1
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 1 or x = 7

Check. When x = 1: left side = 1² − 8 × 1 + 7 = 1 − 8 + 7 = 0. It equals 0. ✓
When x = 7: left side = 7² − 8 × 7 + 7 = 49 − 56 + 7 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² + 5x − 14 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 5 and c = -14.
   We need two numbers that multiply to -14 (the last number) and add to 5 (the middle number). List the pairs that multiply to -14:
   1 × -14 = -14,   add: 1 + -14 = -13 ✗
   -1 × 14 = -14,   add: -1 + 14 = 13 ✗
   2 × -7 = -14,   add: 2 + -7 = -5 ✗
   -2 × 7 = -14,   add: -2 + 7 = 5 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x + 7).
   Check by expanding: x² + 7x − 2x − 14 = x² + 5x − 14. ✓
   x² + 5x − 14 = (x − 2)(x + 7)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(x + 7) = 0
   So (x − 2) = 0 or (x + 7) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 2 = 0
   Add 2 to both sides: x = 2
   
   x + 7 = 0
   Subtract 7 from both sides: x = −7
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −7 or x = 2

Check. When x = −7: left side = (−7)² + 5 × (−7) − 14 = 49 − 35 − 14 = 0. It equals 0. ✓
When x = 2: left side = 2² + 5 × 2 − 14 = 4 + 10 − 14 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   x² − 6x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are x², −6x. The biggest number and the highest power of x that divide into all of them is x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   x² ÷ x = x
   −6x ÷ x = −6
   So x² − 6x = x(x − 6).
   Check by expanding: x × (x − 6) = x × x, x × (−6) → x² − 6x. ✓
   x² − 6x = x(x − 6)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(x − 6) = 0
   So (x − 6) = 0 or x = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 6 = 0
   Add 6 to both sides: x = 6
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 6

Check. When x = 0: left side = 0² − 6 × 0 = 0 + 0 = 0. It equals 0. ✓
When x = 6: left side = 6² − 6 × 6 = 36 − 36 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (2x − 3)(x + 4) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (2x − 3)(x + 4) = 0
   So (x + 4) = 0 or (2x − 3) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x + 4 = 0
   Subtract 4 from both sides: x = −4
   
   2x − 3 = 0
   Add 3 to both sides: 2x = 3
   Divide both sides by 2: x = 3 ÷ 2 = 32
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −4 or x = 32

Check. When x = −4: ((−4) + 4)(2 × (−4) − 3) = 0 × (−11) = 0. ✓
When x = 32: (32 + 4)(2 × 32 − 3) = 112 × 0 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (3x + 1)(x − 5) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (3x + 1)(x − 5) = 0
   So (x − 5) = 0 or (3x + 1) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 5 = 0
   Add 5 to both sides: x = 5
   
   3x + 1 = 0
   Subtract 1 from both sides: 3x = −1
   Divide both sides by 3: x = −1 ÷ 3 = −13
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −13 or x = 5

Check. When x = −13: ((−13) − 5)(3 × (−13) + 1) = (−163) × 0 = 0. ✓
When x = 5: (5 − 5)(3 × 5 + 1) = 0 × 16 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (5x − 2)(2x + 7) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (5x − 2)(2x + 7) = 0
   So (2x + 7) = 0 or (5x − 2) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   2x + 7 = 0
   Subtract 7 from both sides: 2x = −7
   Divide both sides by 2: x = −7 ÷ 2 = −72
   
   5x − 2 = 0
   Add 2 to both sides: 5x = 2
   Divide both sides by 5: x = 2 ÷ 5 = 25
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −72 or x = 25

Check. When x = −72: (2 × (−72) + 7)(5 × (−72) − 2) = 0 × (−392) = 0. ✓
When x = 25: (2 × 25 + 7)(5 × 25 − 2) = 395 × 0 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   x(4x − 3) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   x(4x − 3) = 0
   So x = 0 or (4x − 3) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   4x − 3 = 0
   Add 3 to both sides: 4x = 3
   Divide both sides by 4: x = 3 ÷ 4 = 34
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 34

Check. When x = 0: (0)(4 × 0 − 3) = 0 × (−3) = 0. ✓
When x = 34: (34)(4 × 34 − 3) = 34 × 0 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (2x − 1)² = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (2x − 1)² = 0
   So (2x − 1) = 0.
   A bracket that appears twice gives the same answer twice, so there is only one solution from it (a repeated root).
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   2x − 1 = 0
   Add 1 to both sides: 2x = 1
   Divide both sides by 2: x = 1 ÷ 2 = 12
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 12

Check. When x = 12: (2 × 12 − 1)² = 0² = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. The quadratic is already written as brackets multiplied together and equal to 0.
   (x − 2)(3 − x) = 0
2. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (x − 2)(3 − x) = 0
   So (x − 3) = 0 or (x − 2) = 0.
3. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 3 = 0
   Add 3 to both sides: x = 3
   
   x − 2 = 0
   Add 2 to both sides: x = 2
4. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 2 or x = 3

Check. When x = 2: (2 − 3)(2 − 2) = (−1) × 0 = 0. ✓
When x = 3: (3 − 3)(3 − 2) = 0 × 1 = 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   3x² − 27 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 3x², −27. The biggest number and the highest power of x that divide into all of them is 3.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   3x² ÷ 3 = x²
   −27 ÷ 3 = −9
   So 3x² − 27 = 3(x² − 9).
   Check by expanding: 3 × (x² − 9) = 3 × x², 3 × (−9) → 3x² − 27. ✓
   Can the bracket be factorised again? Look at (x² − 9).
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 9 = 3², so b = 3.
   Put them into the rule: (x − 3)(x + 3).
   Check by expanding: (x − 3)(x + 3) = x² + 3x − 3x − 9 = x² − 9. ✓ (the middle terms cancel)
   3x² − 27 = 3(x − 3)(x + 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   3(x − 3)(x + 3) = 0
   So (x − 3) = 0 or (x + 3) = 0.
   The number 3 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 3 = 0
   Add 3 to both sides: x = 3
   
   x + 3 = 0
   Subtract 3 from both sides: x = −3
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −3 or x = 3

Check. When x = −3: left side = 3 × (−3)² − 27 = 27 − 27 = 0. It equals 0. ✓
When x = 3: left side = 3 × 3² − 27 = 27 − 27 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   2x² − 50 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 2x², −50. The biggest number and the highest power of x that divide into all of them is 2.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   2x² ÷ 2 = x²
   −50 ÷ 2 = −25
   So 2x² − 50 = 2(x² − 25).
   Check by expanding: 2 × (x² − 25) = 2 × x², 2 × (−25) → 2x² − 50. ✓
   Can the bracket be factorised again? Look at (x² − 25).
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. x² = (x)², so a = x. And 25 = 5², so b = 5.
   Put them into the rule: (x − 5)(x + 5).
   Check by expanding: (x − 5)(x + 5) = x² + 5x − 5x − 25 = x² − 25. ✓ (the middle terms cancel)
   2x² − 50 = 2(x − 5)(x + 5)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   2(x − 5)(x + 5) = 0
   So (x − 5) = 0 or (x + 5) = 0.
   The number 2 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 5 = 0
   Add 5 to both sides: x = 5
   
   x + 5 = 0
   Subtract 5 from both sides: x = −5
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −5 or x = 5

Check. When x = −5: left side = 2 × (−5)² − 50 = 50 − 50 = 0. It equals 0. ✓
When x = 5: left side = 2 × 5² − 50 = 50 − 50 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   4x² − 9 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. 4x² = (2x)², so a = 2x. And 9 = 3², so b = 3.
   Put them into the rule: (2x − 3)(2x + 3).
   Check by expanding: (2x − 3)(2x + 3) = 4x² + 32x − 32x − 9 = 4x² − 9. ✓ (the middle terms cancel)
   4x² − 9 = (2x − 3)(2x + 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (2x − 3)(2x + 3) = 0
   So (2x − 3) = 0 or (2x + 3) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   2x − 3 = 0
   Add 3 to both sides: 2x = 3
   Divide both sides by 2: x = 3 ÷ 2 = 32
   
   2x + 3 = 0
   Subtract 3 from both sides: 2x = −3
   Divide both sides by 2: x = −3 ÷ 2 = −32
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −32 or x = 32

Check. When x = −32: left side = 4 × (−32)² − 9 = 9 − 9 = 0. It equals 0. ✓
When x = 32: left side = 4 × 32² − 9 = 9 − 9 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   25x² − 4 = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
   The rule is: a² − b² = (a − b)(a + b).
   Find a and b. 25x² = (5x)², so a = 5x. And 4 = 2², so b = 2.
   Put them into the rule: (5x − 2)(5x + 2).
   Check by expanding: (5x − 2)(5x + 2) = 25x² + 25x − 25x − 4 = 25x² − 4. ✓ (the middle terms cancel)
   25x² − 4 = (5x − 2)(5x + 2)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   (5x − 2)(5x + 2) = 0
   So (5x − 2) = 0 or (5x + 2) = 0.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   5x − 2 = 0
   Add 2 to both sides: 5x = 2
   Divide both sides by 5: x = 2 ÷ 5 = 25
   
   5x + 2 = 0
   Subtract 2 from both sides: 5x = −2
   Divide both sides by 5: x = −2 ÷ 5 = −25
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −25 or x = 25

Check. When x = −25: left side = 25 × (−25)² − 4 = 4 − 4 = 0. It equals 0. ✓
When x = 25: left side = 25 × 25² − 4 = 4 − 4 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   5x² − 20x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 5x², −20x. The biggest number and the highest power of x that divide into all of them is 5x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   5x² ÷ 5x = x
   −20x ÷ 5x = −4
   So 5x² − 20x = 5x(x − 4).
   Check by expanding: 5x × (x − 4) = 5x × x, 5x × (−4) → 5x² − 20x. ✓
   5x² − 20x = 5x(x − 4)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   5x(x − 4) = 0
   So (x − 4) = 0 or x = 0.
   The number 5 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x − 4 = 0
   Add 4 to both sides: x = 4
   
   x = 0
   x = 0 is a solution.
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = 0 or x = 4

Check. When x = 0: left side = 5 × 0² − 20 × 0 = 0 + 0 = 0. It equals 0. ✓
When x = 4: left side = 5 × 4² − 20 × 4 = 80 − 80 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.

Key words. A quadratic equation is an equation whose highest power of the letter is 2 (it has an x² term), like x² + 5x + 6 = 0. To solve it means to find every value of x that makes it true. These values are called the solutions or roots. To factorise means to write it as brackets multiplied together.

Method. To solve a quadratic equation by factorising: (1) get one side equal to 0, (2) factorise, (3) use the null factor law to make each bracket equal to 0, (4) solve the little equations.

1. Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
   6x² + 9x = 0
2. Check that one side is 0. We can only use the null factor law when one side of the equation is exactly 0.
   The right-hand side is already 0, so there is nothing to move.
3. Factorise the quadratic. Factorising writes it as brackets multiplied together. Then the null factor law can be used.
   Look for a common factor. A common factor is something that divides into every term.
   The terms are 6x², 9x. The biggest number and the highest power of x that divide into all of them is 3x.
   Take it outside a bracket, and inside the bracket put each term divided by it:
   6x² ÷ 3x = 2x
   9x ÷ 3x = 3
   So 6x² + 9x = 3x(2x + 3).
   Check by expanding: 3x × (2x + 3) = 3x × 2x, 3x × 3 → 6x² + 9x. ✓
   6x² + 9x = 3x(2x + 3)
4. Use the null factor law. The null factor law. If two numbers multiply to make 0, then at least one of them must be 0. Think about it: 7 × 0 = 0 and 0 × 12 = 0, but 7 × 12 is not 0. The same is true when the numbers are brackets: if (x + 2)(x − 5) = 0, then either the first bracket is 0 or the second bracket is 0.
   3x(2x + 3) = 0
   So x = 0 or (2x + 3) = 0.
   The number 3 in front can never be 0, so it gives no solution. Ignore it.
5. Solve each bracket separately. Each bracket gives a very small equation of the kind you solved in Chapter 9. Use inverse operations on both sides.
   x = 0
   x = 0 is a solution.
   
   2x + 3 = 0
   Subtract 3 from both sides: 2x = −3
   Divide both sides by 2: x = −3 ÷ 2 = −32
6. Write the solutions. Quadratics usually have two solutions (a curve crosses the x-axis twice).
   x = −32 or x = 0

Check. When x = −32: left side = 6 × (−32)² + 9 × (−32) = 272 − 272 = 0. It equals 0. ✓
When x = 0: left side = 6 × 0² + 9 × 0 = 0 + 0 = 0. It equals 0. ✓

Watch out. Do not divide both sides by x: you would lose the solution x = 0. And never set a bracket equal to a number other than 0.