GCSE Maths · Quadratic and Simultaneous Equations · Higher · Lesson 14.9

GCSE Maths: The quadratic formula (Higher)

Part of the GCSE Maths quadratic and simultaneous equations chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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14.9

The quadratic formula

Edexcel 1MA1: A18
Grade 6-8

Many quadratic equations cannot be factorised with whole numbers. For example x² + 3x − 5 = 0: you need two whole numbers that multiply to −5 and add to 3, and there are none. Yet the equation does have solutions (the graph crosses the axis). The quadratic formula finds them for any quadratic equation.

The formula. If ax² + bx + c = 0 then

x = −b ± √(b² − 4ac)2a

The ± sign means “plus or minus”: do the calculation once with + and once with −, giving two solutions. The number b² − 4ac is called the discriminant.

Where does it come from? It is just completing the square (section 14.10) done once for all quadratics. Start with ax² + bx + c = 0. Divide by a: x² + (b/a)x + c/a = 0. Complete the square: (x + b/2a)² − b²/4a² + c/a = 0. Move the numbers across: (x + b/2a)² = (b² − 4ac)/4a². Take the square root of both sides (remembering ±): x + b/2a = ±√(b² − 4ac)/2a. Subtract b/2a and you have the formula. So the formula is a short-cut that always works.

Using it. (1) Rearrange so that one side is 0 and the equation is in the form ax² + bx + c = 0. (2) Write down a, b and c with their signs. (3) Substitute into the formula, using brackets for negative numbers. (4) Work out b² − 4ac first. (5) Take the square root and finish. The answer may be a whole number, a fraction, a decimal rounded as the question asks, or an exact surd such as 3 ± √5 (see Chapter 6 for simplifying surds).

The discriminant tells you how many solutions. If b² − 4ac is positive there are two solutions. If it is 0 there is exactly one (a repeated root). If it is negative there are no real solutions because you cannot take the square root of a negative number: the graph never crosses the x-axis.

Key ideas

Quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a.
First rearrange to equal 0. Write down a, b, c with their signs and substitute in brackets.
The discriminant b² − 4ac: positive gives 2 solutions, zero gives 1, negative gives none.
Exact answers keep the surd, e.g. 3 ± √5. Decimal answers are rounded only at the very end.
The whole of the top, −b ± √…, is divided by 2a.

Worked example 14.9.1

Solve 2x² + 5x − 3 = 0 using the quadratic formula.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
2x² + 5x − 3 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² + 5x − 3 = 0
a = 2, b = 5, c = −3
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −5 ± √(5² − 4 × 2 × (−3))2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 5² = 25
4ac = 4 × 2 × (−3) = −24
b² − 4ac = 25 − (−24) = 49
−b = −5
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −5 ± √494
Find the square root of 49. 7 × 7 = 49, so √49 = 7. The ± sign means we do one calculation with + and one with −.
x = −5 ± 74
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: −5 + 74 = 24 = 12
With −: −5 − 74 = −124 = −3

Check. When x = −3: 2 × (−3)² + 5 × (−3) − 3 = 18 − 15 − 3 = 0. ✓
When x = 12: 2 × 12² + 5 × 12 − 3 = 12 + 52 − 3 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Worked example 14.9.2

Solve x² − 6x + 4 = 0. Give your answers in the form p ± √q.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² − 6x + 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − 6x + 4 = 0
a = 1, b = −6, c = 4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−6) ± √((−6)² − 4 × 1 × 4)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−6)² = 36
4ac = 4 × 1 × 4 = 16
b² − 4ac = 36 − 16 = 20
−b = −(−6) = 6
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 6 ± √202
Simplify √20. 20 = 4 × 5 and 4 is a square number, so √20 = √4 × √5 = 2√5. (Look for the biggest square number that divides into 20.)
√20 = 2√5
Cancel the common factor in the fraction. Every number in the fraction (6, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = 6 ± 2√52 = 3 ± √5

Check. Substitute x = 0.76: 1 × (0.76)² + (−6) × (0.76) + 4 ≈ 0.018, which is very close to 0 (it is not exactly 0 only because 0.76 is rounded). ✓
Substitute x = 5.24: 1 × (5.24)² + (−6) × (5.24) + 4 ≈ 0.018, which is very close to 0 (it is not exactly 0 only because 5.24 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Worked example 14.9.3

Solve 3x² + 4x − 2 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
3x² + 4x − 2 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
3x² + 4x − 2 = 0
a = 3, b = 4, c = −2
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −4 ± √(4² − 4 × 3 × (−2))2 × 3
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 4² = 16
4ac = 4 × 3 × (−2) = −24
b² − 4ac = 16 − (−24) = 40
−b = −4
2a = 2 × 3 = 6
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −4 ± √406
Simplify √40. 40 = 4 × 10 and 4 is a square number, so √40 = √4 × √10 = 2√10. (Look for the biggest square number that divides into 40.)
√40 = 2√10
Cancel the common factor in the fraction. Every number in the fraction (4, 2 and 6) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −4 ± 2√106 = −2 ± √103
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −4 + 2√106 = 0.3874… ≈ 0.39
With −: x = −4 − 2√106 = −1.7208… ≈ −1.72

Check. Substitute x = −1.72: 3 × (−1.72)² + 4 × (−1.72) + (−2) ≈ −0.005, which is very close to 0 (it is not exactly 0 only because −1.72 is rounded). ✓
Substitute x = 0.39: 3 × (0.39)² + 4 × (0.39) + (−2) ≈ 0.016, which is very close to 0 (it is not exactly 0 only because 0.39 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Worked example 14.9.4

Solve x² + 3x = 5. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

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 = 5
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 5 from both sides: x² + 3x − 5 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 3x − 5 = 0
a = 1, b = 3, c = −5
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −3 ± √(3² − 4 × 1 × (−5))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 3² = 9
4ac = 4 × 1 × (−5) = −20
b² − 4ac = 9 − (−20) = 29
−b = −3
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −3 ± √292
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = −3 ± √292
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −3 + √292 = 1.1926… ≈ 1.19
With −: x = −3 − √292 = −4.1926… ≈ −4.19

Check. Substitute x = −4.19: 1 × (−4.19)² + 3 × (−4.19) + (−5) ≈ −0.014, which is very close to 0 (it is not exactly 0 only because −4.19 is rounded). ✓
Substitute x = 1.19: 1 × (1.19)² + 3 × (1.19) + (−5) ≈ −0.014, which is very close to 0 (it is not exactly 0 only because 1.19 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Worked example 14.9.5

Show that x² + 2x + 5 = 0 has no real solutions.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² + 2x + 5 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 2x + 5 = 0
a = 1, b = 2, c = 5
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −2 ± √(2² − 4 × 1 × 5)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 2² = 4
4ac = 4 × 1 × 5 = 20
b² − 4ac = 4 − 20 = −16
−b = −2
2a = 2 × 1 = 2
Look at the discriminant. You cannot take the square root of a negative number and get an ordinary (real) number, because a number multiplied by itself is never negative.
−16 is negative, so there are no real solutions.

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Watch out

The most common slips are: (1) forgetting that −b is positive when b is negative, (2) writing −5² = −25 instead of (−5)² = 25, (3) dividing only the square root by 2a instead of the whole top line, and (4) forgetting the b² − 4ac sign change when c is negative: 4ac is negative so subtracting it adds.

Exercise 14I The quadratic formula

Calculator allowed

FSolve using the quadratic formula. These solutions are rational (whole numbers or fractions).

(a)2x² − x − 6 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
2x² − x − 6 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² − x − 6 = 0
a = 2, b = −1, c = −6
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−1) ± √((−1)² − 4 × 2 × (−6))2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−1)² = 1
4ac = 4 × 2 × (−6) = −48
b² − 4ac = 1 − (−48) = 49
−b = −(−1) = 1
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 1 ± √494
Find the square root of 49. 7 × 7 = 49, so √49 = 7. The ± sign means we do one calculation with + and one with −.
x = 1 ± 74
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: 1 + 74 = 84 = 2
With −: 1 − 74 = −64 = −32

Check. When x = −32: 2 × (−32)² − (−32) − 6 = 92 + 32 − 6 = 0. ✓
When x = 2: 2 × 2² − 2 − 6 = 8 − 2 − 6 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(b)3x² + x − 4 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
3x² + x − 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
3x² + x − 4 = 0
a = 3, b = 1, c = −4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −1 ± √(1² − 4 × 3 × (−4))2 × 3
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 1² = 1
4ac = 4 × 3 × (−4) = −48
b² − 4ac = 1 − (−48) = 49
−b = −1
2a = 2 × 3 = 6
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −1 ± √496
Find the square root of 49. 7 × 7 = 49, so √49 = 7. The ± sign means we do one calculation with + and one with −.
x = −1 ± 76
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: −1 + 76 = 66 = 1
With −: −1 − 76 = −86 = −43

Check. When x = −43: 3 × (−43)² + (−43) − 4 = 163 − 43 − 4 = 0. ✓
When x = 1: 3 × 1² + 1 − 4 = 3 + 1 − 4 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(c)x² − 4x − 12 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² − 4x − 12 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − 4x − 12 = 0
a = 1, b = −4, c = −12
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−4) ± √((−4)² − 4 × 1 × (−12))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−4)² = 16
4ac = 4 × 1 × (−12) = −48
b² − 4ac = 16 − (−48) = 64
−b = −(−4) = 4
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 4 ± √642
Find the square root of 64. 8 × 8 = 64, so √64 = 8. The ± sign means we do one calculation with + and one with −.
x = 4 ± 82
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: 4 + 82 = 122 = 6
With −: 4 − 82 = −42 = −2

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

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(d)2x² + 7x + 3 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
2x² + 7x + 3 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² + 7x + 3 = 0
a = 2, b = 7, c = 3
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −7 ± √(7² − 4 × 2 × 3)2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 7² = 49
4ac = 4 × 2 × 3 = 24
b² − 4ac = 49 − 24 = 25
−b = −7
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −7 ± √254
Find the square root of 25. 5 × 5 = 25, so √25 = 5. The ± sign means we do one calculation with + and one with −.
x = −7 ± 54
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: −7 + 54 = −24 = −12
With −: −7 − 54 = −124 = −3

Check. When x = −3: 2 × (−3)² + 7 × (−3) + 3 = 18 − 21 + 3 = 0. ✓
When x = −12: 2 × (−12)² + 7 × (−12) + 3 = 12 − 72 + 3 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(e)5x² + 4x − 1 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
5x² + 4x − 1 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
5x² + 4x − 1 = 0
a = 5, b = 4, c = −1
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −4 ± √(4² − 4 × 5 × (−1))2 × 5
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 4² = 16
4ac = 4 × 5 × (−1) = −20
b² − 4ac = 16 − (−20) = 36
−b = −4
2a = 2 × 5 = 10
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −4 ± √3610
Find the square root of 36. 6 × 6 = 36, so √36 = 6. The ± sign means we do one calculation with + and one with −.
x = −4 ± 610
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: −4 + 610 = 210 = 15
With −: −4 − 610 = −1010 = −1

Check. When x = −1: 5 × (−1)² + 4 × (−1) − 1 = 5 − 4 − 1 = 0. ✓
When x = 15: 5 × 15² + 4 × 15 − 1 = 15 + 45 − 1 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(f)6x² − 5x − 4 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
6x² − 5x − 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
6x² − 5x − 4 = 0
a = 6, b = −5, c = −4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−5) ± √((−5)² − 4 × 6 × (−4))2 × 6
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−5)² = 25
4ac = 4 × 6 × (−4) = −96
b² − 4ac = 25 − (−96) = 121
−b = −(−5) = 5
2a = 2 × 6 = 12
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 5 ± √12112
Find the square root of 121. 11 × 11 = 121, so √121 = 11. The ± sign means we do one calculation with + and one with −.
x = 5 ± 1112
Work out the two answers. One calculation with the + sign and one with the − sign. Simplify each fraction.
With +: 5 + 1112 = 1612 = 43
With −: 5 − 1112 = −612 = −12

Check. When x = −12: 6 × (−12)² − 5 × (−12) − 4 = 32 + 52 − 4 = 0. ✓
When x = 43: 6 × 43² − 5 × 43 − 4 = 323 − 203 − 4 = 0. ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve x² + 5x + 2 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² + 5x + 2 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 5x + 2 = 0
a = 1, b = 5, c = 2
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −5 ± √(5² − 4 × 1 × 2)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 5² = 25
4ac = 4 × 1 × 2 = 8
b² − 4ac = 25 − 8 = 17
−b = −5
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −5 ± √172
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = −5 ± √172
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −5 + √172 = −0.4384… ≈ −0.44
With −: x = −5 − √172 = −4.5616… ≈ −4.56

Check. Substitute x = −4.56: 1 × (−4.56)² + 5 × (−4.56) + 2 ≈ −0.006, which is very close to 0 (it is not exactly 0 only because −4.56 is rounded). ✓
Substitute x = −0.44: 1 × (−0.44)² + 5 × (−0.44) + 2 ≈ −0.006, which is very close to 0 (it is not exactly 0 only because −0.44 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve x² − 3x − 5 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² − 3x − 5 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − 3x − 5 = 0
a = 1, b = −3, c = −5
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−3) ± √((−3)² − 4 × 1 × (−5))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−3)² = 9
4ac = 4 × 1 × (−5) = −20
b² − 4ac = 9 − (−20) = 29
−b = −(−3) = 3
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 3 ± √292
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = 3 ± √292
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = 3 + √292 = 4.1926… ≈ 4.19
With −: x = 3 − √292 = −1.1926… ≈ −1.19

Check. Substitute x = −1.19: 1 × (−1.19)² + (−3) × (−1.19) + (−5) ≈ −0.014, which is very close to 0 (it is not exactly 0 only because −1.19 is rounded). ✓
Substitute x = 4.19: 1 × (4.19)² + (−3) × (4.19) + (−5) ≈ −0.014, which is very close to 0 (it is not exactly 0 only because 4.19 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve 2x² + 3x − 4 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
2x² + 3x − 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² + 3x − 4 = 0
a = 2, b = 3, c = −4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −3 ± √(3² − 4 × 2 × (−4))2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 3² = 9
4ac = 4 × 2 × (−4) = −32
b² − 4ac = 9 − (−32) = 41
−b = −3
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −3 ± √414
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = −3 ± √414
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −3 + √414 = 0.8508… ≈ 0.85
With −: x = −3 − √414 = −2.3508… ≈ −2.35

Check. Substitute x = −2.35: 2 × (−2.35)² + 3 × (−2.35) + (−4) ≈ −0.005, which is very close to 0 (it is not exactly 0 only because −2.35 is rounded). ✓
Substitute x = 0.85: 2 × (0.85)² + 3 × (0.85) + (−4) ≈ −0.005, which is very close to 0 (it is not exactly 0 only because 0.85 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve 3x² − 8x + 2 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
3x² − 8x + 2 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
3x² − 8x + 2 = 0
a = 3, b = −8, c = 2
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−8) ± √((−8)² − 4 × 3 × 2)2 × 3
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−8)² = 64
4ac = 4 × 3 × 2 = 24
b² − 4ac = 64 − 24 = 40
−b = −(−8) = 8
2a = 2 × 3 = 6
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 8 ± √406
Simplify √40. 40 = 4 × 10 and 4 is a square number, so √40 = √4 × √10 = 2√10. (Look for the biggest square number that divides into 40.)
√40 = 2√10
Cancel the common factor in the fraction. Every number in the fraction (8, 2 and 6) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = 8 ± 2√106 = 4 ± √103
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = 8 + 2√106 = 2.3874… ≈ 2.39
With −: x = 8 − 2√106 = 0.2792… ≈ 0.28

Check. Substitute x = 0.28: 3 × (0.28)² + (−8) × (0.28) + 2 ≈ −0.005, which is very close to 0 (it is not exactly 0 only because 0.28 is rounded). ✓
Substitute x = 2.39: 3 × (2.39)² + (−8) × (2.39) + 2 ≈ 0.016, which is very close to 0 (it is not exactly 0 only because 2.39 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve 4x² − 7x + 1 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
4x² − 7x + 1 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
4x² − 7x + 1 = 0
a = 4, b = −7, c = 1
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−7) ± √((−7)² − 4 × 4 × 1)2 × 4
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−7)² = 49
4ac = 4 × 4 × 1 = 16
b² − 4ac = 49 − 16 = 33
−b = −(−7) = 7
2a = 2 × 4 = 8
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 7 ± √338
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = 7 ± √338
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = 7 + √338 = 1.5931… ≈ 1.59
With −: x = 7 − √338 = 0.1569… ≈ 0.16

Check. Substitute x = 0.16: 4 × (0.16)² + (−7) × (0.16) + 1 ≈ −0.018, which is very close to 0 (it is not exactly 0 only because 0.16 is rounded). ✓
Substitute x = 1.59: 4 × (1.59)² + (−7) × (1.59) + 1 ≈ −0.018, which is very close to 0 (it is not exactly 0 only because 1.59 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve 5x² + 2x − 4 = 0. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
5x² + 2x − 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
5x² + 2x − 4 = 0
a = 5, b = 2, c = −4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −2 ± √(2² − 4 × 5 × (−4))2 × 5
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 2² = 4
4ac = 4 × 5 × (−4) = −80
b² − 4ac = 4 − (−80) = 84
−b = −2
2a = 2 × 5 = 10
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −2 ± √8410
Simplify √84. 84 = 4 × 21 and 4 is a square number, so √84 = √4 × √21 = 2√21. (Look for the biggest square number that divides into 84.)
√84 = 2√21
Cancel the common factor in the fraction. Every number in the fraction (2, 2 and 10) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −2 ± 2√2110 = −1 ± √215
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −2 + 2√2110 = 0.7165… ≈ 0.72
With −: x = −2 − 2√2110 = −1.1165… ≈ −1.12

Check. Substitute x = −1.12: 5 × (−1.12)² + 2 × (−1.12) + (−4) ≈ 0.032, which is very close to 0 (it is not exactly 0 only because −1.12 is rounded). ✓
Substitute x = 0.72: 5 × (0.72)² + 2 × (0.72) + (−4) ≈ 0.032, which is very close to 0 (it is not exactly 0 only because 0.72 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FSolve, giving your answers in the form p ± √q or as exact fractions.

(a)x² + 4x + 1 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² + 4x + 1 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 4x + 1 = 0
a = 1, b = 4, c = 1
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −4 ± √(4² − 4 × 1 × 1)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 4² = 16
4ac = 4 × 1 × 1 = 4
b² − 4ac = 16 − 4 = 12
−b = −4
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −4 ± √122
Simplify √12. 12 = 4 × 3 and 4 is a square number, so √12 = √4 × √3 = 2√3. (Look for the biggest square number that divides into 12.)
√12 = 2√3
Cancel the common factor in the fraction. Every number in the fraction (4, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −4 ± 2√32 = −2 ± √3

Check. Substitute x = −3.73: 1 × (−3.73)² + 4 × (−3.73) + 1 ≈ −0.007, which is very close to 0 (it is not exactly 0 only because −3.73 is rounded). ✓
Substitute x = −0.27: 1 × (−0.27)² + 4 × (−0.27) + 1 ≈ −0.007, which is very close to 0 (it is not exactly 0 only because −0.27 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(b)x² − 2x − 7 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² − 2x − 7 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − 2x − 7 = 0
a = 1, b = −2, c = −7
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−2) ± √((−2)² − 4 × 1 × (−7))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−2)² = 4
4ac = 4 × 1 × (−7) = −28
b² − 4ac = 4 − (−28) = 32
−b = −(−2) = 2
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 2 ± √322
Simplify √32. 32 = 16 × 2 and 16 is a square number, so √32 = √16 × √2 = 4√2. (Look for the biggest square number that divides into 32.)
√32 = 4√2
Cancel the common factor in the fraction. Every number in the fraction (2, 4 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = 2 ± 4√22 = 1 ± 2√2

Check. Substitute x = −1.83: 1 × (−1.83)² + (−2) × (−1.83) + (−7) ≈ 0.009, which is very close to 0 (it is not exactly 0 only because −1.83 is rounded). ✓
Substitute x = 3.83: 1 × (3.83)² + (−2) × (3.83) + (−7) ≈ 0.009, which is very close to 0 (it is not exactly 0 only because 3.83 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(c)x² + 6x + 2 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² + 6x + 2 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 6x + 2 = 0
a = 1, b = 6, c = 2
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −6 ± √(6² − 4 × 1 × 2)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 6² = 36
4ac = 4 × 1 × 2 = 8
b² − 4ac = 36 − 8 = 28
−b = −6
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −6 ± √282
Simplify √28. 28 = 4 × 7 and 4 is a square number, so √28 = √4 × √7 = 2√7. (Look for the biggest square number that divides into 28.)
√28 = 2√7
Cancel the common factor in the fraction. Every number in the fraction (6, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −6 ± 2√72 = −3 ± √7

Check. Substitute x = −5.65: 1 × (−5.65)² + 6 × (−5.65) + 2 ≈ 0.023, which is very close to 0 (it is not exactly 0 only because −5.65 is rounded). ✓
Substitute x = −0.35: 1 × (−0.35)² + 6 × (−0.35) + 2 ≈ 0.023, which is very close to 0 (it is not exactly 0 only because −0.35 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(d)x² − 8x + 9 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² − 8x + 9 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − 8x + 9 = 0
a = 1, b = −8, c = 9
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−8) ± √((−8)² − 4 × 1 × 9)2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−8)² = 64
4ac = 4 × 1 × 9 = 36
b² − 4ac = 64 − 36 = 28
−b = −(−8) = 8
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 8 ± √282
Simplify √28. 28 = 4 × 7 and 4 is a square number, so √28 = √4 × √7 = 2√7. (Look for the biggest square number that divides into 28.)
√28 = 2√7
Cancel the common factor in the fraction. Every number in the fraction (8, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = 8 ± 2√72 = 4 ± √7

Check. Substitute x = 1.35: 1 × (1.35)² + (−8) × (1.35) + 9 ≈ 0.023, which is very close to 0 (it is not exactly 0 only because 1.35 is rounded). ✓
Substitute x = 6.65: 1 × (6.65)² + (−8) × (6.65) + 9 ≈ 0.023, which is very close to 0 (it is not exactly 0 only because 6.65 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(e)2x² + 6x + 3 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
2x² + 6x + 3 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² + 6x + 3 = 0
a = 2, b = 6, c = 3
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −6 ± √(6² − 4 × 2 × 3)2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 6² = 36
4ac = 4 × 2 × 3 = 24
b² − 4ac = 36 − 24 = 12
−b = −6
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −6 ± √124
Simplify √12. 12 = 4 × 3 and 4 is a square number, so √12 = √4 × √3 = 2√3. (Look for the biggest square number that divides into 12.)
√12 = 2√3
Cancel the common factor in the fraction. Every number in the fraction (6, 2 and 4) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −6 ± 2√34 = −3 ± √32

Check. Substitute x = −2.37: 2 × (−2.37)² + 6 × (−2.37) + 3 ≈ 0.014, which is very close to 0 (it is not exactly 0 only because −2.37 is rounded). ✓
Substitute x = −0.63: 2 × (−0.63)² + 6 × (−0.63) + 3 ≈ 0.014, which is very close to 0 (it is not exactly 0 only because −0.63 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

(f)3x² − 6x − 1 = 0

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
3x² − 6x − 1 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
3x² − 6x − 1 = 0
a = 3, b = −6, c = −1
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−6) ± √((−6)² − 4 × 3 × (−1))2 × 3
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−6)² = 36
4ac = 4 × 3 × (−1) = −12
b² − 4ac = 36 − (−12) = 48
−b = −(−6) = 6
2a = 2 × 3 = 6
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 6 ± √486
Simplify √48. 48 = 16 × 3 and 16 is a square number, so √48 = √16 × √3 = 4√3. (Look for the biggest square number that divides into 48.)
√48 = 4√3
Cancel the common factor in the fraction. Every number in the fraction (6, 4 and 6) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = 6 ± 4√36 = 3 ± 2√33

Check. Substitute x = −0.15: 3 × (−0.15)² + (−6) × (−0.15) + (−1) ≈ −0.033, which is very close to 0 (it is not exactly 0 only because −0.15 is rounded). ✓
Substitute x = 2.15: 3 × (2.15)² + (−6) × (2.15) + (−1) ≈ −0.033, which is very close to 0 (it is not exactly 0 only because 2.15 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FRearrange and solve x² + 3x = 5. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

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 = 5
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 5 from both sides: x² + 3x − 5 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 3x − 5 = 0
a = 1, b = 3, c = −5
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −3 ± √(3² − 4 × 1 × (−5))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 3² = 9
4ac = 4 × 1 × (−5) = −20
b² − 4ac = 9 − (−20) = 29
−b = −3
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −3 ± √292
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = −3 ± √292
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −3 + √292 = 1.1926… ≈ 1.19
With −: x = −3 − √292 = −4.1926… ≈ −4.19

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FExpand, rearrange and solve x(x + 4) = 7. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
x(x + 4) = 7
Expand any brackets first. Multiply every term in one bracket by every term in the other. We cannot see how to factorise until the brackets are gone.
Expand x(x + 4):
x × x = x²
x × 4 = 4x
so x(x + 4) = x² + 4x
The equation is now: x² + 4x = 7
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 7 from both sides: x² + 4x − 7 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 4x − 7 = 0
a = 1, b = 4, c = −7
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −4 ± √(4² − 4 × 1 × (−7))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 4² = 16
4ac = 4 × 1 × (−7) = −28
b² − 4ac = 16 − (−28) = 44
−b = −4
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −4 ± √442
Simplify √44. 44 = 4 × 11 and 4 is a square number, so √44 = √4 × √11 = 2√11. (Look for the biggest square number that divides into 44.)
√44 = 2√11
Cancel the common factor in the fraction. Every number in the fraction (4, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −4 ± 2√112 = −2 ± √11
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −4 + 2√112 = 1.3166… ≈ 1.32
With −: x = −4 − 2√112 = −5.3166… ≈ −5.32

Check. Substitute x = −5.32: 1 × (−5.32)² + 4 × (−5.32) + (−7) ≈ 0.022, which is very close to 0 (it is not exactly 0 only because −5.32 is rounded). ✓
Substitute x = 1.32: 1 × (1.32)² + 4 × (1.32) + (−7) ≈ 0.022, which is very close to 0 (it is not exactly 0 only because 1.32 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FRearrange and solve 2x² = 5x + 4. Give your answers correct to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

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 = 5x + 4
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 5x from both sides: 2x² − 5x = 4
Subtract 4 from both sides: 2x² − 5x − 4 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² − 5x − 4 = 0
a = 2, b = −5, c = −4
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−5) ± √((−5)² − 4 × 2 × (−4))2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−5)² = 25
4ac = 4 × 2 × (−4) = −32
b² − 4ac = 25 − (−32) = 57
−b = −(−5) = 5
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 5 ± √574
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = 5 ± √574
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = 5 + √574 = 3.1375… ≈ 3.14
With −: x = 5 − √574 = −0.6375… ≈ −0.64

Check. Substitute x = −0.64: 2 × (−0.64)² + (−5) × (−0.64) + (−4) ≈ 0.019, which is very close to 0 (it is not exactly 0 only because −0.64 is rounded). ✓
Substitute x = 3.14: 2 × (3.14)² + (−5) × (3.14) + (−4) ≈ 0.019, which is very close to 0 (it is not exactly 0 only because 3.14 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

FExpand, rearrange and solve exactly (x + 2)(x − 3) = 5. Give your answers in the form p ± √q or as exact fractions.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. A quadratic equation contains a squared letter. We want every value of the letter that makes the two sides equal.
(x + 2)(x − 3) = 5
Expand any brackets first. Multiply every term in one bracket by every term in the other. We cannot see how to factorise until the brackets are gone.
Expand (x + 2)(x − 3):
First: x × x = x²
Outer: x × (−3) = −3x
Inner: 2 × x = 2x
Last: 2 × (−3) = −6
so (x + 2)(x − 3) = x² − x − 6
The equation is now: x² − x − 6 = 5
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 5 from both sides: x² − x − 6 − 5 = 0
Collect like terms: x² − x − 11 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² − x − 11 = 0
a = 1, b = −1, c = −11
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−1) ± √((−1)² − 4 × 1 × (−11))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−1)² = 1
4ac = 4 × 1 × (−11) = −44
b² − 4ac = 1 − (−44) = 45
−b = −(−1) = 1
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 1 ± √452
Simplify √45. 45 = 9 × 5 and 9 is a square number, so √45 = √9 × √5 = 3√5. (Look for the biggest square number that divides into 45.)
√45 = 3√5
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = 1 ± 3√52

Check. Substitute x = −2.85: 1 × (−2.85)² + (−1) × (−2.85) + (−11) ≈ −0.028, which is very close to 0 (it is not exactly 0 only because −2.85 is rounded). ✓
Substitute x = 3.85: 1 × (3.85)² + (−1) × (3.85) + (−11) ≈ −0.028, which is very close to 0 (it is not exactly 0 only because 3.85 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

RUse the discriminant to decide how many solutions each equation has. You do not need to solve them.

(a)x² + x + 1 = 0

(b)x² − 4x + 4 = 0

(c)x² − 3x − 1 = 0

(d)2x² + 3x + 5 = 0

Done

PSThe height, h metres, of a ball t seconds after it is thrown is given by h = 12t − 5t². Find the times when the ball is 6 m above the ground. Give your answers to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The height is 6 metres, so put h = 6 in the formula.
12t − 5t² = 6
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Add 5t² to both sides: 12t = 6 + 5t²
Subtract 12t from both sides: 0 = 6 + 5t² − 12t
Collect like terms: 0 = 5t² − 12t + 6
Write it the other way round: 5t² − 12t + 6 = 0
Read off a, b and c. a is the number in front of t², b is the number in front of t, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
5t² − 12t + 6 = 0
a = 5, b = −12, c = 6
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
t = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
t = −(−12) ± √((−12)² − 4 × 5 × 6)2 × 5
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−12)² = 144
4ac = 4 × 5 × 6 = 120
b² − 4ac = 144 − 120 = 24
−b = −(−12) = 12
2a = 2 × 5 = 10
Put the pieces into the formula. Now the formula only needs a square root and a division.
t = 12 ± √2410
Simplify √24. 24 = 4 × 6 and 4 is a square number, so √24 = √4 × √6 = 2√6. (Look for the biggest square number that divides into 24.)
√24 = 2√6
Cancel the common factor in the fraction. Every number in the fraction (12, 2 and 10) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
t = 12 ± 2√610 = 6 ± √65
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: t = 12 + 2√610 = 1.6899… ≈ 1.69
With −: t = 12 − 2√610 = 0.7101… ≈ 0.71
Answer the question. Both times are positive, so both are possible: the ball is at 6 m on the way up and again on the way down.
The ball is 6 m high after 0.71 seconds (going up) and after 1.69 seconds (coming down).

Check. Substitute t = 0.71: 5 × (0.71)² + (−12) × (0.71) + 6 ≈ 0.001, which is very close to 0 (it is not exactly 0 only because 0.71 is rounded). ✓
Substitute t = 1.69: 5 × (1.69)² + (−12) × (1.69) + 6 ≈ 0.001, which is very close to 0 (it is not exactly 0 only because 1.69 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

PSA rectangle has width x cm and length (x + 3) cm. Its area is 20 cm². Find the width and the length, to 2 decimal places.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Form the equation. Let the width be x cm. The length is (x + 3) cm. Area = length × width = 20.
x(x + 3) = 20
Expand any brackets first. Multiply every term in one bracket by every term in the other. We cannot see how to factorise until the brackets are gone.
Expand x(x + 3):
x × x = x²
x × 3 = 3x
so x(x + 3) = x² + 3x
The equation is now: x² + 3x = 20
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Subtract 20 from both sides: x² + 3x − 20 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 3x − 20 = 0
a = 1, b = 3, c = −20
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −3 ± √(3² − 4 × 1 × (−20))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 3² = 9
4ac = 4 × 1 × (−20) = −80
b² − 4ac = 9 − (−80) = 89
−b = −3
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −3 ± √892
Write the exact answer. An answer with a square root left in it is called an exact answer (a surd). It is exactly right, with no rounding.
x = −3 ± √892
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = −3 + √892 = 3.2170… ≈ 3.22
With −: x = −3 − √892 = −6.2170… ≈ −6.22
Decide which solution makes sense and find the length. A width cannot be negative, so reject x = −6.22. The length is x + 3.
Width = 3.22 cm, length = 3.22 + 3 = 6.22 cm
The rectangle is 3.22 cm by 6.22 cm.

Check. Substitute x = −6.22: 1 × (−6.22)² + 3 × (−6.22) + (−20) ≈ 0.028, which is very close to 0 (it is not exactly 0 only because −6.22 is rounded). ✓
Substitute x = 3.22: 1 × (3.22)² + 3 × (3.22) + (−20) ≈ 0.028, which is very close to 0 (it is not exactly 0 only because 3.22 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

PSA farmer has 40 m of fencing to fence three sides of a rectangular field. The fourth side is a wall. The two equal sides are each x m long and the area of the field is 160 m². Find the possible values of x to 2 decimal places, and the possible dimensions.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Form the equation. A farmer has 40 m of fence for three sides of a rectangle (the fourth side is a wall). Let the two equal sides be x m. The third side is 40 − 2x m. Area = 160 m².
x(40 − 2x) = 160
Expand any brackets first. Multiply every term in one bracket by every term in the other. We cannot see how to factorise until the brackets are gone.
Expand x(−2x + 40):
x × (−2x) = −2x²
x × 40 = 40x
so x(−2x + 40) = −2x² + 40x
The equation is now: −2x² + 40x = 160
Rearrange so that one side equals 0. A quadratic must be rearranged so that one side is exactly 0. The null factor law (see below) only works on a product that equals 0. Move every term to the side where the squared term is positive, using inverse operations on both sides.
Add 2x² to both sides: 40x = 160 + 2x²
Subtract 40x from both sides: 0 = 160 + 2x² − 40x
Collect like terms: 0 = 2x² − 40x + 160
Write it the other way round: 2x² − 40x + 160 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
2x² − 40x + 160 = 0
a = 2, b = −40, c = 160
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −(−40) ± √((−40)² − 4 × 2 × 160)2 × 2
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = (−40)² = 1600
4ac = 4 × 2 × 160 = 1280
b² − 4ac = 1600 − 1280 = 320
−b = −(−40) = 40
2a = 2 × 2 = 4
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = 40 ± √3204
Simplify √320. 320 = 64 × 5 and 64 is a square number, so √320 = √64 × √5 = 8√5. (Look for the biggest square number that divides into 320.)
√320 = 8√5
Cancel the common factor in the fraction. Every number in the fraction (40, 8 and 4) can be divided by 4. Divide the top and the bottom by 4. You can only cancel a factor that divides every term on the top.
x = 40 ± 8√54 = 10 ± 2√5
Use a calculator and round to 2 decimal places. Type the whole top of the fraction in brackets, then divide by the bottom. Do the + and the − separately. Do not round until the end.
With +: x = 40 + 8√54 = 14.4721… ≈ 14.47
With −: x = 40 − 8√54 = 5.5279… ≈ 5.53
Decide which solutions make sense. Both values of x are positive, and the third side 40 − 2x is positive for both, so both are possible.
x = 5.53: third side = 40 − 11.06 = 28.94 m
x = 14.47: third side = 40 − 28.94 = 11.06 m
There are two possible rectangles: 5.53 m by 28.94 m, or 14.47 m by 11.06 m.

Check. Substitute x = 5.53: 2 × (5.53)² + (−40) × (5.53) + 160 ≈ −0.038, which is very close to 0 (it is not exactly 0 only because 5.53 is rounded). ✓
Substitute x = 14.47: 2 × (14.47)² + (−40) × (14.47) + 160 ≈ −0.038, which is very close to 0 (it is not exactly 0 only because 14.47 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

RUse the quadratic formula to show that the solutions of x² + 2x − 1 = 0 are x = −1 ± √2.

Key words. The discriminant is b² − 4ac. A surd is a square root that is not a whole number, like √5. An exact answer keeps the surd; a decimal answer is rounded.

Method. Rearrange to ax² + bx + c = 0, read off a, b, c, substitute them (in brackets) into the quadratic formula, work out the discriminant first, then give the answer in the form asked.

Write down the equation. The quadratic formula solves any equation in the form ax² + bx + c = 0.
x² + 2x − 1 = 0
Read off a, b and c. a is the number in front of x², b is the number in front of x, and c is the number on its own. Each number keeps its sign (a minus sign belongs to the number after it). If there is no number in front of a letter, it is 1.
x² + 2x − 1 = 0
a = 1, b = 2, c = −1
Write down the quadratic formula. This formula works for every quadratic equation. (It is proved by completing the square in general; see the start of this section.)
x = −b ± √(b² − 4ac)2a
Substitute a, b and c, using brackets. Brackets stop sign mistakes, especially when a number is negative, because (−5)² = 25 but −5² would be −25.
x = −2 ± √(2² − 4 × 1 × (−1))2 × 1
Work out the pieces one at a time. The part under the square root, b² − 4ac, is called the discriminant. Work it out separately and carefully; it decides what kind of answer we get.
b² = 2² = 4
4ac = 4 × 1 × (−1) = −4
b² − 4ac = 4 − (−4) = 8
−b = −2
2a = 2 × 1 = 2
Put the pieces into the formula. Now the formula only needs a square root and a division.
x = −2 ± √82
Simplify √8. 8 = 4 × 2 and 4 is a square number, so √8 = √4 × √2 = 2√2. (Look for the biggest square number that divides into 8.)
√8 = 2√2
Cancel the common factor in the fraction. Every number in the fraction (2, 2 and 2) can be divided by 2. Divide the top and the bottom by 2. You can only cancel a factor that divides every term on the top.
x = −2 ± 2√22 = −1 ± √2

Check. Substitute x = −2.41: 1 × (−2.41)² + 2 × (−2.41) + (−1) ≈ −0.012, which is very close to 0 (it is not exactly 0 only because −2.41 is rounded). ✓
Substitute x = 0.41: 1 × (0.41)² + 2 × (0.41) + (−1) ≈ −0.012, which is very close to 0 (it is not exactly 0 only because 0.41 is rounded). ✓

Watch out. Put every number in brackets when you substitute. The whole top line is divided by 2a, not just the square root.

Done

RFind all the values of k for which x² + kx + 9 = 0 has no real solutions.

Done