GCSE Maths: Expanding single brackets (Higher)

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 4(x + 3) means 4 multiplied by everything inside the bracket, not just the first term.
   4(x + 3) = 4 × x + 4 × 3
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   4 × x = 4x
   4 × 3 = 12
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   4x + 12

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 20 and the answer gives 20. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 5(y − 2) means 5 multiplied by everything inside the bracket, not just the first term.
   5(y − 2): 5 is multiplied by each of the terms +y, −2
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   5 × y = 5y
   5 × (−2) = −10
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   5y − 10

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 2(3a + 1) means 2 multiplied by everything inside the bracket, not just the first term.
   2(3a + 1) = 2 × 3a + 2 × 1
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   2 × 3a = 6a
   2 × 1 = 2
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   6a + 2

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives 14 and the answer gives 14. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 7(2 − b) means 7 multiplied by everything inside the bracket, not just the first term.
   7(2 − b): 7 is multiplied by each of the terms +2, −b
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   7 × 2 = 14
   7 × (−b) = −7b
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −7b + 14

Check. Test with numbers. Put b = 2 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 6(2p + 3q) means 6 multiplied by everything inside the bracket, not just the first term.
   6(2p + 3q) = 6 × 2p + 6 × 3q
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   6 × 2p = 12p
   6 × 3q = 18q
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   12p + 18q

Check. Test with numbers. Put p = 2, q = 3 into the original and into your answer. The original gives 78 and the answer gives 78. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 9(1 + 2c) means 9 multiplied by everything inside the bracket, not just the first term.
   9(1 + 2c) = 9 × 1 + 9 × 2c
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   9 × 1 = 9
   9 × 2c = 18c
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   18c + 9

Check. Test with numbers. Put c = 2 into the original and into your answer. The original gives 45 and the answer gives 45. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So x(x + 5) means x multiplied by everything inside the bracket, not just the first term.
   x(x + 5) = x × x + x × 5
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   x × x = x²
   x × 5 = 5x
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   x² + 5x

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 14 and the answer gives 14. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So y(y − 3) means y multiplied by everything inside the bracket, not just the first term.
   y(y − 3): y is multiplied by each of the terms +y, −3
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   y × y = y²
   y × (−3) = −3y
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   y² − 3y

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives −2 and the answer gives −2. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So a(2a + 1) means a multiplied by everything inside the bracket, not just the first term.
   a(2a + 1) = a × 2a + a × 1
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   a × 2a = 2a²
   a × 1 = a
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   2a² + a

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives 10 and the answer gives 10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So m(4 − m) means m multiplied by everything inside the bracket, not just the first term.
   m(4 − m): m is multiplied by each of the terms +4, −m
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   m × 4 = 4m
   m × (−m) = −m²
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −m² + 4m

Check. Test with numbers. Put m = 2 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 3x(x + 2) means 3x multiplied by everything inside the bracket, not just the first term.
   3x(x + 2) = 3x × x + 3x × 2
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   3x × x = 3x²
   3x × 2 = 6x
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   3x² + 6x

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 24 and the answer gives 24. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 2a(a − 4) means 2a multiplied by everything inside the bracket, not just the first term.
   2a(a − 4): 2a is multiplied by each of the terms +a, −4
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   2a × a = 2a²
   2a × (−4) = −8a
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   2a² − 8a

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives −8 and the answer gives −8. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 5y(2y + 3) means 5y multiplied by everything inside the bracket, not just the first term.
   5y(2y + 3) = 5y × 2y + 5y × 3
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   5y × 2y = 10y²
   5y × 3 = 15y
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   10y² + 15y

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives 70 and the answer gives 70. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 4m(3m − 1) means 4m multiplied by everything inside the bracket, not just the first term.
   4m(3m − 1): 4m is multiplied by each of the terms +3m, −1
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   4m × 3m = 12m²
   4m × (−1) = −4m
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   12m² − 4m

Check. Test with numbers. Put m = 2 into the original and into your answer. The original gives 40 and the answer gives 40. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So −2(x + 3) means −2 multiplied by everything inside the bracket, not just the first term.
   −2(x + 3): (−2) is multiplied by each of the terms +x, +3
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   (−2) × x = −2x
   (−2) × 3 = −6
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −2x − 6

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives −10 and the answer gives −10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So −5(a − 1) means −5 multiplied by everything inside the bracket, not just the first term.
   −5(a − 1): (−5) is multiplied by each of the terms +a, −1
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   (−5) × a = −5a
   (−5) × (−1) = 5
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −5a + 5

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives −5 and the answer gives −5. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So −x(x − 4) means −x multiplied by everything inside the bracket, not just the first term.
   −x(x − 4): (−x) is multiplied by each of the terms +x, −4
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   (−x) × x = −x²
   (−x) × (−4) = 4x
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −x² + 4x

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So −3(2y − 5) means −3 multiplied by everything inside the bracket, not just the first term.
   −3(2y − 5): (−3) is multiplied by each of the terms +2y, −5
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   (−3) × 2y = −6y
   (−3) × (−5) = 15
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −6y + 15

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives 3 and the answer gives 3. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   3(x + 1) + 2(x + 4)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 3(x + 1)
   3 × x = 3x
   3 × 1 = 3
   
   Bracket: 2(x + 4)
   2 × x = 2x
   2 × 4 = 8
3. Write all the pieces in one line. Now there are no brackets, only terms.
   3x + 3 + 2x + 8
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x terms: +3x, +2x
      add the coefficients: 3 + 2 = 5  so they give 5x
   number terms: +3, +8
      add the coefficients: 3 + 8 = 11  so they give 11
   Answer: 5x + 11

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 21 and the answer gives 21. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   4(a + 2) + 3(a − 1)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 4(a + 2)
   4 × a = 4a
   4 × 2 = 8
   
   Bracket: 3(a − 1)
   3 × a = 3a
   3 × (−1) = −3
3. Write all the pieces in one line. Now there are no brackets, only terms.
   4a + 8 + 3a − 3
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   a terms: +4a, +3a
      add the coefficients: 4 + 3 = 7  so they give 7a
   number terms: +8, −3
      add the coefficients: 8 − 3 = 5  so they give 5
   Answer: 7a + 5

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives 19 and the answer gives 19. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   5(y + 3) + 2(y − 6)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 5(y + 3)
   5 × y = 5y
   5 × 3 = 15
   
   Bracket: 2(y − 6)
   2 × y = 2y
   2 × (−6) = −12
3. Write all the pieces in one line. Now there are no brackets, only terms.
   5y + 15 + 2y − 12
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   y terms: +5y, +2y
      add the coefficients: 5 + 2 = 7  so they give 7y
   number terms: +15, −12
      add the coefficients: 15 − 12 = 3  so they give 3
   Answer: 7y + 3

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives 17 and the answer gives 17. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   2(3x + 1) + 5(x + 2)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 2(3x + 1)
   2 × 3x = 6x
   2 × 1 = 2
   
   Bracket: 5(x + 2)
   5 × x = 5x
   5 × 2 = 10
3. Write all the pieces in one line. Now there are no brackets, only terms.
   6x + 2 + 5x + 10
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x terms: +6x, +5x
      add the coefficients: 6 + 5 = 11  so they give 11x
   number terms: +2, +10
      add the coefficients: 2 + 10 = 12  so they give 12
   Answer: 11x + 12

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 34 and the answer gives 34. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   5(x + 2) − 3(x + 1)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 5(x + 2)
   5 × x = 5x
   5 × 2 = 10
   
   Bracket: −3(x + 1)
   (−3) × x = −3x
   (−3) × 1 = −3
3. Write all the pieces in one line. Now there are no brackets, only terms.
   5x + 10 − 3x − 3
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x terms: +5x, −3x
      add the coefficients: 5 − 3 = 2  so they give 2x
   number terms: +10, −3
      add the coefficients: 10 − 3 = 7  so they give 7
   Answer: 2x + 7

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   4(a + 5) − 2(a + 3)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 4(a + 5)
   4 × a = 4a
   4 × 5 = 20
   
   Bracket: −2(a + 3)
   (−2) × a = −2a
   (−2) × 3 = −6
3. Write all the pieces in one line. Now there are no brackets, only terms.
   4a + 20 − 2a − 6
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   a terms: +4a, −2a
      add the coefficients: 4 − 2 = 2  so they give 2a
   number terms: +20, −6
      add the coefficients: 20 − 6 = 14  so they give 14
   Answer: 2a + 14

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives 18 and the answer gives 18. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   7(y + 1) − 4(y − 2)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 7(y + 1)
   7 × y = 7y
   7 × 1 = 7
   
   Bracket: −4(y − 2)
   (−4) × y = −4y
   (−4) × (−2) = 8
3. Write all the pieces in one line. Now there are no brackets, only terms.
   7y + 7 − 4y + 8
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   y terms: +7y, −4y
      add the coefficients: 7 − 4 = 3  so they give 3y
   number terms: +7, +8
      add the coefficients: 7 + 8 = 15  so they give 15
   Answer: 3y + 15

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives 21 and the answer gives 21. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   3(2m + 1) − 2(m − 4)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 3(2m + 1)
   3 × 2m = 6m
   3 × 1 = 3
   
   Bracket: −2(m − 4)
   (−2) × m = −2m
   (−2) × (−4) = 8
3. Write all the pieces in one line. Now there are no brackets, only terms.
   6m + 3 − 2m + 8
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   m terms: +6m, −2m
      add the coefficients: 6 − 2 = 4  so they give 4m
   number terms: +3, +8
      add the coefficients: 3 + 8 = 11  so they give 11
   Answer: 4m + 11

Check. Test with numbers. Put m = 2 into the original and into your answer. The original gives 19 and the answer gives 19. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   x(x + 4) − 2(x − 3)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: x(x + 4)
   x × x = x²
   x × 4 = 4x
   
   Bracket: −2(x − 3)
   (−2) × x = −2x
   (−2) × (−3) = 6
3. Write all the pieces in one line. Now there are no brackets, only terms.
   x² + 4x − 2x + 6
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +4x, −2x
      add the coefficients: 4 − 2 = 2  so they give 2x
   number terms: +6  (only one of this type, so it stays as it is)
   Answer: x² + 2x + 6

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 14 and the answer gives 14. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   2a(a + 1) + 3(a − 5)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 2a(a + 1)
   2a × a = 2a²
   2a × 1 = 2a
   
   Bracket: 3(a − 5)
   3 × a = 3a
   3 × (−5) = −15
3. Write all the pieces in one line. Now there are no brackets, only terms.
   2a² + 2a + 3a − 15
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   a² terms: +2a²  (only one of this type, so it stays as it is)
   a terms: +2a, +3a
      add the coefficients: 2 + 3 = 5  so they give 5a
   number terms: −15  (only one of this type, so it stays as it is)
   Answer: 2a² + 5a − 15

Check. Test with numbers. Put a = 2 into the original and into your answer. The original gives 3 and the answer gives 3. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   y(2y − 1) − y(y + 3)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: y(2y − 1)
   y × 2y = 2y²
   y × (−1) = −y
   
   Bracket: −y(y + 3)
   (−y) × y = −y²
   (−y) × 3 = −3y
3. Write all the pieces in one line. Now there are no brackets, only terms.
   2y² − y − y² − 3y
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   y² terms: +2y², −y²
      add the coefficients: 2 − 1 = 1  so they give y²
   y terms: −y, −3y
      add the coefficients: −1 − 3 = −4  so they give −4y
   Answer: y² − 4y

Check. Test with numbers. Put y = 2 into the original and into your answer. The original gives −4 and the answer gives −4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 2xy(3x − 4y) means 2xy multiplied by everything inside the bracket, not just the first term.
   2xy(3x − 4y): 2xy is multiplied by each of the terms +3x, −4y
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   2xy × 3x = 6x²y
   2xy × (−4y) = −8xy²
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   6x²y − 8xy²

Check. Test with numbers. Put x = 2, y = 3 into the original and into your answer. The original gives −72 and the answer gives −72. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So 3ab(a + 2b) means 3ab multiplied by everything inside the bracket, not just the first term.
   3ab(a + 2b) = 3ab × a + 3ab × 2b
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). All the signs are positive here, so every answer is positive.
   3ab × a = 3a²b
   3ab × 2b = 6ab²
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   3a²b + 6ab²

Check. Test with numbers. Put a = 2, b = 3 into the original and into your answer. The original gives 144 and the answer gives 144. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So −2p(p² − pq + 3q²) means −2p multiplied by everything inside the bracket, not just the first term.
   −2p(p² − pq + 3q²): (−2p) is multiplied by each of the terms +p², −pq, +3q²
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   (−2p) × p² = −2p³
   (−2p) × (−pq) = 2p²q
   (−2p) × 3q² = −6pq²
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   −2p³ + 2p²q − 6pq²

Check. Test with numbers. Put p = 2, q = 3 into the original and into your answer. The original gives −100 and the answer gives −100. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Key words. To expand a bracket means to multiply it out so that the brackets disappear. The term outside the bracket is multiplied by each term inside.

Method. To expand a bracket, multiply every term inside the bracket by the term outside it.

1. Say what the bracket means. A number or letter written next to a bracket means multiply. So x²(x − y) means x² multiplied by everything inside the bracket, not just the first term.
   x²(x − y): x² is multiplied by each of the terms +x, −y
2. Multiply the outside term by each term inside, one at a time. Multiply the numbers first, then the letters (add the powers of any letter that appears in both). Watch the signs: Use the sign rules: a positive times a positive is positive, a negative times a negative is positive, and a positive times a negative (either way round) is negative.
   x² × x = x³
   x² × (−y) = −x²y
3. Put the answers together. The results join up as a single expression. Each keeps its own sign.
   x³ − x²y

Check. Test with numbers. Put x = 2, y = 3 into the original and into your answer. The original gives −4 and the answer gives −4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Multiply every term in the bracket, including the last one. And be careful with minus signs: −3(x − 2) = −3x + 6, because −3 × −2 = +6.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   2x(3x − 1) − x(x − 4)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 2x(3x − 1)
   2x × 3x = 6x²
   2x × (−1) = −2x
   
   Bracket: −x(x − 4)
   (−x) × x = −x²
   (−x) × (−4) = 4x
3. Write all the pieces in one line. Now there are no brackets, only terms.
   6x² − 2x − x² + 4x
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x² terms: +6x², −x²
      add the coefficients: 6 − 1 = 5  so they give 5x²
   x terms: −2x, +4x
      add the coefficients: −2 + 4 = 2  so they give 2x
   Answer: 5x² + 2x

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 24 and the answer gives 24. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   3a(2a − b) + 2b(a + 3b)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front.
   Bracket: 3a(2a − b)
   3a × 2a = 6a²
   3a × (−b) = −3ab
   
   Bracket: 2b(a + 3b)
   2b × a = 2ab
   2b × 3b = 6b²
3. Write all the pieces in one line. Now there are no brackets, only terms.
   6a² − 3ab + 2ab + 6b²
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   a² terms: +6a²  (only one of this type, so it stays as it is)
   ab terms: −3ab, +2ab
      add the coefficients: −3 + 2 = −1  so they give −ab
   b² terms: +6b²  (only one of this type, so it stays as it is)
   Answer: 6a² − ab + 6b²

Check. Test with numbers. Put a = 2, b = 3 into the original and into your answer. The original gives 72 and the answer gives 72. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. Expand every bracket first (keeping each sign with its term), then collect the like terms.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   4(x² + 2x) − x(3x − 5)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 4(x² + 2x)
   4 × x² = 4x²
   4 × 2x = 8x
   
   Bracket: −x(3x − 5)
   (−x) × 3x = −3x²
   (−x) × (−5) = 5x
3. Write all the pieces in one line. Now there are no brackets, only terms.
   4x² + 8x − 3x² + 5x
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x² terms: +4x², −3x²
      add the coefficients: 4 − 3 = 1  so they give x²
   x terms: +8x, +5x
      add the coefficients: 8 + 5 = 13  so they give 13x
   Answer: x² + 13x

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 30 and the answer gives 30. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. A minus sign in front of a bracket changes the sign of every term inside it. −2(x − 5) = −2x + 10. Many mistakes are made with the last term.

Method. To show that two expressions are equal, work on the more complicated one (expand the brackets and collect like terms) until it becomes the other one.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   3(2x + 1) − 2(3x − 4)
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: 3(2x + 1)
   3 × 2x = 6x
   3 × 1 = 3
   
   Bracket: −2(3x − 4)
   (−2) × 3x = −6x
   (−2) × (−4) = 8
3. Write all the pieces in one line. Now there are no brackets, only terms.
   6x + 3 − 6x + 8
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x terms: +6x, −6x
      add the coefficients: 6 − 6 = 0  so they cancel to give 0
   number terms: +3, +8
      add the coefficients: 3 + 8 = 11  so they give 11
   Answer: 11
5. Compare with what we were asked to show. In a "show that" question the last line must be exactly the expression in the question, followed by a short sentence.
   11 is the same as 11, so the statement is shown to be true. ✓

Check. Test with numbers. Put into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Show every line of working. The last line must match the target exactly, and finish with a sentence.

Method. To show that two expressions are equal, work on the more complicated one (expand the brackets and collect like terms) until it becomes the other one.

1. Write the expression. There is more than one bracket. We expand each bracket separately, and then collect like terms.
   x(x + 5) − 5(x − 2) − x²
2. Expand each bracket on its own. Multiply every term in the bracket by the term in front of it, including the sign in front. A minus in front of a bracket means you multiply by a negative number, so every sign inside the bracket flips.
   Bracket: x(x + 5)
   x × x = x²
   x × 5 = 5x
   
   Bracket: −5(x − 2)
   (−5) × x = −5x
   (−5) × (−2) = 10
   
   −x² has no bracket, so it stays as it is.
3. Write all the pieces in one line. Now there are no brackets, only terms.
   x² + 5x − 5x + 10 − x²
4. Collect the like terms. Group the terms with the same letters and powers and add their coefficients.
   x² terms: +x², −x²
      add the coefficients: 1 − 1 = 0  so they cancel to give 0
   x terms: +5x, −5x
      add the coefficients: 5 − 5 = 0  so they cancel to give 0
   number terms: +10  (only one of this type, so it stays as it is)
   Answer: 10
5. Compare with what we were asked to show. In a "show that" question the last line must be exactly the expression in the question, followed by a short sentence.
   10 is the same as 10, so the statement is shown to be true. ✓

Check. Test with numbers. Put x = 4 into the original and into your answer. The original gives 10 and the answer gives 10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Show every line of working. The last line must match the target exactly, and finish with a sentence.

Method. Divide each term of the answer by the number outside.

1. Look at the constant term. After expanding, the number term is 3 × ? = 21.
   3 × ? = 21
2. Solve. Divide 21 by 3.
   ? = 21 ÷ 3 = 7

Check. 3(x + 7) = 3x + 21. ✓

Method. The number outside multiplies every term. Compare the x terms.

1. Look at the x terms. ? × 2x = 8x.
   ? × 2 = 8
2. Solve.
   ? = 8 ÷ 2 = 4

Check. 4(2x − 5) = 8x − 20. ✓ (4 × −5 = −20 as well.)

Method. The outside term x multiplies the first term to give x².

1. Look at the first term. x × ? = x².
   x × ? = x²
2. Solve. Divide x² by x: subtract the powers.
   ? = x² ÷ x = x

Check. x(x + 7) = x² + 7x. ✓