Revision mat · The Quadratic Formula

16 marks

Question mix weighted by 26 real exam questions from 26 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

The quadratic formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$Setup $ax^2+bx+c=0$Discriminant $\Delta=b^2-4ac$Nature of roots $b^2-4ac \;\begin{cases}>0\\=0\\<0\end{cases}$
Using the formula≈77% of real exam Qs
1Solve $x^2 + 6x - 4 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.
Answer: (3 marks)
2Solve $4x^2 - 2x - 1 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.
Answer: (3 marks)
3Solve $x^2 + 9x + 3 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.
Answer: (3 marks)
4Solve $3x^2 + 4x - 6 = 0$ using the quadratic formula.
Give your answers in surd form.
Answer: (3 marks)
★ Exam capstonemixed & other forms ≈23% of real exam Qs
5Solve $3x^2 = -8x + 7$.
Give your answers correct to 2 decimal places.
Answer: (4 marks)

Mark scheme · The Quadratic Formula

Total: 16 marks

Award the marks shown for each correct step, then add up the total out of 16. A method mark counts even if the final answer is wrong.

1Solve $x^2 + 6x - 4 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.[3 marks]
Method
$a = 1,\ b = 6,\ c = -4$
$x = \dfrac{-6 \pm \sqrt{36 + 16}}{2} = \dfrac{-6 \pm \sqrt{52}}{2}$
$\sqrt{52} = 7.211\ldots$
So $x = 0.61$ or $x = -6.61$ (2 d.p.)
Answer: $x = 0.61 or x = -6.61$
Marks
✔1 markSubstitute correctly into the formula
✔1 markDiscriminant = 52 and √ evaluated
✔1 markBoth roots to 2 d.p.: 0.61 and -6.61
2Solve $4x^2 - 2x - 1 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.[3 marks]
Method
$a = 4,\ b = -2,\ c = -1$
$x = \dfrac{2 \pm \sqrt{4 + 16}}{8} = \dfrac{2 \pm \sqrt{20}}{8}$
$\sqrt{20} = 4.472\ldots$
So $x = 0.81$ or $x = -0.31$ (2 d.p.)
Answer: $x = 0.81 or x = -0.31$
Marks
✔1 markSubstitute correctly into the formula
✔1 markDiscriminant = 20 and √ evaluated
✔1 markBoth roots to 2 d.p.: 0.81 and -0.31
3Solve $x^2 + 9x + 3 = 0$ using the quadratic formula.
Give your answers correct to 2 decimal places.[3 marks]
Method
$a = 1,\ b = 9,\ c = 3$
$x = \dfrac{-9 \pm \sqrt{81 - 12}}{2} = \dfrac{-9 \pm \sqrt{69}}{2}$
$\sqrt{69} = 8.307\ldots$
So $x = -0.35$ or $x = -8.65$ (2 d.p.)
Answer: $x = -0.35 or x = -8.65$
Marks
✔1 markSubstitute correctly into the formula
✔1 markDiscriminant = 69 and √ evaluated
✔1 markBoth roots to 2 d.p.: -0.35 and -8.65
4Solve $3x^2 + 4x - 6 = 0$ using the quadratic formula.
Give your answers in surd form.[3 marks]
Method
$a = 3,\ b = 4,\ c = -6$
Discriminant: $b^2 - 4ac = 16 + 72 = 88$
$x = \dfrac{-4 \pm \sqrt{88}}{6} = \dfrac{-2 \pm \sqrt{22}}{3}$
Answer: $$x = \dfrac{-2 \pm \sqrt{22}}{3}$$
Marks
✔1 markSubstitute a, b, c into the formula
✔1 markDiscriminant b² − 4ac = 88
✔1 markSimplify the surd and give both roots
5Solve $3x^2 = -8x + 7$.
Give your answers correct to 2 decimal places.[4 marks]
Method
Rearrange to $3x^2 + 8x - 7 = 0$
$a = 3,\ b = 8,\ c = -7$
$x = \dfrac{-8 \pm \sqrt{64 + 84}}{6} = \dfrac{-8 \pm \sqrt{148}}{6}$
So $x = 0.69$ or $x = -3.36$ (2 d.p.)
Answer: $x = 0.69 or x = -3.36$
Marks
✔1 markRearrange to 3x^2 + 8x - 7 = 0
✔1 markSubstitute correctly into the formula
✔1 markDiscriminant = 148 and √ evaluated
✔1 markBoth roots to 2 d.p.: 0.69 and -3.36
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