The quadratic formula Quiz (Grade 7)

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Score: 0 / 10

The quadratic formula — Notes & Worked Examples Show / Hide

Explanation:

Use the quadratic formula x = −b ± √(b² − 4ac)2a when a quadratic equation ax² + bx + c = 0 cannot be easily factorised, or when the question asks for solutions to a given number of decimal places or significant figures, or in surd form.

Always rearrange the equation so that the right-hand side equals 0 first, then write down the values of a, b, and c including their signs. Put negative numbers in brackets when substituting into (−b)² − 4(a)(c) so your calculator squares the negative sign properly.

Example 1: Solve 2x² − 7x − 3 = 0 to 2 d.p.

Solution:

Here a = 2, b = −7, c = −3

x = −(−7) ± √((−7)² − 4(2)(−3))2(2)

x = 7 ± √(49 + 24)4 = 7 ± √734

x = 3.8860... or x = −0.3860...

Answer: x = 3.89 or x = −0.39 (2 d.p.)

Example 2: Solve x² + 6x = 4 in the form p ± √q.

Solution:

Rearrange to x² + 6x − 4 = 0 (a = 1, b = 6, c = −4)

x = −6 ± √(6² − 4(1)(−4))2(1) = −6 ± √522

Since √52 = √4 × √13 = 2√13:

x = −6 ± 2√132 = −3 ± √13

Answer: x = −3 ± √13

One Page Guide (PDF) The quadratic formula Worksheets

Question 1 AQA Jun 2025 2H Q15

[2 marks]

Solve 3x² + 5x - 9 = 0

Give your solutions as decimals.

Worked Solution & Answer

x = [–5 ± √(5² – 4(3)(–9))] / (2 × 3) = [–5 ± √133] / 6. M1
x ≈ 1.09 and x ≈ –2.76. A1

Answer: 1.09 and –2.76

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Question 2 AQA Nov 2022 2H Q19

[2 marks]

Using the quadratic formula, or otherwise, solve 3x² + x - 5 = 0

Answer

Worked Solution & Answer

x = (–1 ± √(1² – 4(3)(–5))) / (2 × 3) = (–1 ± √61) / 6. M1
x = 1.14 or x = –1.47 (to 2 d.p.). A1

Answer: Answer 1.14 and -1.47

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Question 3 AQA Jun 2022 2H Q18

[2 marks]

Solve x² + 7x - 11 = 0
Give your solutions as decimals.

Answer

Worked Solution & Answer

Use quadratic formula with a = 1, b = 7, c = -11: M1
x = -7 ± √(7² - 4(1)(-11))2(1) = -7 ± √(49 + 44)2 = -7 ± √(93)2
√(93) ≈ 9.64365
x₁ = -7 + 9.643652 ≈ 1.32
x₂ = -7 - 9.643652 ≈ -8.32 A1

Answer: Answer 1.32 and -8.32

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Question 4 OCR Nov 2020 4H Q15

[3 marks]

(b) @@TABULAR₀@@
Describe the error and write the correct solution.

(b) @@TABULAR₀@@
Describe the error and write the correct solution. [3 marks]

Worked Solution & Answer

(b)

Error: −b was not divided by 2a M2
Correct solution: x = (8 ± √(64 − 24)) / 4 = (8 ± √40) / 4 ⇒ x = 0.419 or x = 3.58 (3 s.f.). C1

Answer (b): Error: The −(−8) term should also be divided by 2 × 2 (the fraction line should go under the whole numerator).
Correct solution: x = (8 ± √40) / 4 ⇒ x = 0.419 or x = 3.58.

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Question 5 AQA Jun 2019 3H Q18

[3 marks]

(a) Write x(3x - 9) = 4 in the form ax² + bx + c = 0 where a, b and c are integers.
18 (b) Solve x(3x - 9) = 4
Give your answers to 2 decimal places.

(a) Write x(3x - 9) = 4 in the form ax² + bx + c = 0 where a, b and c are integers. [1 mark]

Answer

(b) Solve x(3x - 9) = 4
Give your answers to 2 decimal places. [2 marks]

Answer

Worked Solution & Answer

(a)

Expand the bracket: 3x² - 9x = 4
Rearrange to equal 0: 3x² - 9x - 4 = 0 B1

Answer (a): Answer 3x² - 9x - 4 = 0

(b)

Using the quadratic formula for 3x² - 9x - 4 = 0: M1
x = -(-9) ± √((-9)² - 4(3)(-4))2(3) = 9 ± √(81 + 48)6 = 9 ± √(129)6
x = 9 + 11.35786 = 3.39
x = 9 - 11.35786 = -0.39 A1

Answer (b): Answer 3.39 and –0.39

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Question 6 OCR Nov 2017 4H Q18

[4 marks]

Solve this equation algebraically.
Give your solutions correct to 2 decimal places.
3x² + 5x - 1 = 0

x =

Worked Solution & Answer

Using the quadratic formula with a = 3, b = 5, c = −1: M2
x = (−5 ± √(5² − 4 × 3 × (−1))) / (2 × 3) = (−5 ± √(25 + 12)) / 6 = (−5 ± √37) / 6. M1
x = (−5 + 6.08276) / 6 = 0.1804... ≈ 0.18, or x = (−5 − 6.08276) / 6 = −1.8471... ≈ −1.85. A1

Answer: x = 0.18 or -1.85

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Question 7 AQA Nov 2020 2H Q29

[5 marks]

Solve 54x + 1 = 2xx² + 3
Give your solutions to 3 significant figures.
You must show your working.

Work out the values of x.

x =

Worked Solution & Answer

Cross-multiply: 5(x² + 3) = 2x(4x + 1) M1
5x² + 15 = 8x² + 2x ⇒ 3x² + 2x - 15 = 0 M1
Quadratic formula: x = -2 ± √(2² - 4(3)(-15))2(3) = -2 ± √(4 + 180)6 = -2 ± √(184)6 M1
√(184) ≈ 13.5647 M1
x = -2 + 13.56476 ≈ 1.927 ⇒ 1.93
x = -2 - 13.56476 ≈ -2.594 ⇒ -2.59. A1

Answer: x = 1.93 or -2.59

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Question 8 IGCSE Nov 2023 2H Q24

[5 marks]

Solve 45x³ - 80x3x² + x - 4 × (13x - 4 + 1x) = 4(x + 2)5x - 8
Show clear algebraic working.

x =

Worked Solution & Answer

45x³ − 80x = 5x(9x² − 16) = 5x(3x − 4)(3x + 4). 3x² + x − 4 = (3x + 4)(x − 1) M2
Inside bracket: (x + 3x − 4)/(x(3x − 4)) = 4(x − 1)/(x(3x − 4)) M1
Product on LHS = [5x(3x − 4)(3x + 4) / ((3x + 4)(x − 1))] × [4(x − 1) / (x(3x − 4))] = 20 M1
So 20 = 4(x + 2)/(5x − 8) ⇒ 5(5x − 8) = x + 2 ⇒ 25x − 40 = x + 2 ⇒ 24x = 42 ⇒ x = 7/4 (or 1.75). A1

Answer: x = 74

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Question 9

[7 marks]

(a)(i) Write x² – 8x + 3 in the form (x – a)² – b (2) (ii) Write down the turning point (1) (b) Solve 7x² + 8x – 5 = 0 to 3 s.f. (3) (c) What mistake has Alex made in (3k – 2)(k + 4) = 0 ⇒ k = 2 or –4? (1)

(a) Write x² - 8x + 3 in the form (x - a)² - b where a and b are integers. [2 marks]

(b) Hence, write down the coordinates of the turning point on the graph of y = x² - 8x + 3 [1 mark]

(b) Solve 7x² + 8x - 5 = 0
Give your solutions correct to 3 significant figures. [3 marks]

Worked Solution & Answer

(a)

(x – 4)² – 16 + 3 = (x – 4)² – 13 M1
⇒ TP (4, –13) A1

Answer (a): (x - 4)² - 13

(b)

4, –13

Answer (b): 4, –13

(b)

x = (–8 ± √(64 + 140)) / 14 = (–8 ± √204) / 14 = 0.4487... and –1.5916 M2 A1

Answer (b): 0.449 and –1.59

(c)

From 3k – 2 = 0, k = 2/3 C1

Answer (c): He forgot to divide 2 by 3 when solving 3k – 2 = 0 (k should be 2/3, not 2)

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Question 10

[8 marks]

(a) Expand and simplify (x – 2)(2x + 3)(x + 1) (3) (b) (y⁴ × yⁿ)/y² = y⁻³. Find n. (2) (c) Solve 5x² – 4x – 3 = 0 to 3 s.f. (3)

(a) Expand and simplify (x - 2)(2x + 3)(x + 1) [3 marks]

(b) Find the value of n. [2 marks]

Worked Solution & Answer

(a)

(2x² – x – 6)(x + 1) = 2x³ + x² – 7x – 6 M2 A1

Answer (a): 2x³ + x² - 7x - 6

(b)

4 + n – 2 = –3 M1
⇒ n = –5 A1

Answer (b): –5

(c)

x = [4 ± √(16 – 4(5)(–3))] / 10 = (4 ± √76) / 10 M2
⇒ x = 1.27 or –0.472 A1

Answer (c): 1.27, –0.472

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Score: 0 / 10 (0 / 41 marks)

The quadratic formula Worksheets Next: Factorising harder quadratics →

Grade 7 Quizzes

19 Topics

7A - Surds	7B - Bounds
7C - Proportion (harder)	7D - Algebraic fractions
7E - The quadratic formula	7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs	7H - Rearranging (harder)
7J - Functions (harder)	7K - Iteration
7L - Area of any triangle	7M - The sine rule
7N - The cosine rule	7P - Congruent triangles
7Q - 3D pythagoras / trig.	7R - Histograms
7S - Conditional probability	7T - Exact trig. values (harder)
7U - Area of a segment