Quadratic equations: factorising, formula & completing the square – IGCSE Maths exam questions with mark scheme

1Grade 4–5HigherCalculator4 marks

(a) Factorise $x^2 + 3x - 28$ [2]

(b) Solve $2x^2 - 5x - 12 = 0$ [2]

(a) for $(x \pm 7)(x \pm 4)$ or $(x + a)(x + b)$ where $ab = -28$ or $a + b = 3$M1
$(x + 7)(x - 4)$A1
(b) for $(2x + 3)(x - 4)$ or correct substitution into the quadratic formulaM1
$x = -\dfrac{3}{2}$ and $x = 4$A1
Answer: (a) $(x + 7)(x - 4)$ (b) $x = -1.5$, $x = 4$
Examiner tip: In (b) you must go on from the brackets to the solutions — factorising alone gets only the method mark.

2Grade 6–7HigherCalculator3 marks

Solve $3x^2 + 4x - 9 = 0$

Show your working clearly.

Give your solutions correct to 3 significant figures. [3]

for $\dfrac{-4 \pm \sqrt{4^2 - 4 \times 3 \times (-9)}}{2 \times 3}$ (allow one sign error in substitution, but not in the formula), or a correct first step in completing the square, e.g. $3\left[\left(x + \dfrac{2}{3}\right)^2 - \dfrac{4}{9}\right] - 9 = 0$M1
for $\dfrac{-4 \pm \sqrt{124}}{6}$ oe, e.g. $-\dfrac{2}{3} \pm \sqrt{\dfrac{31}{9}}$M1
awrt $1.19$ and awrt $-2.52$A1
Answer: $x = 1.19$ or $x = -2.52$
Examiner tip: With c = −9, −4ac = +108; dropping the negative sign under the square root is the classic error.

3Grade 6–7HigherCalculator5 marks

The diagram shows a rectangle.

All measurements are in centimetres.

The area of the rectangle is 60 cm$^2$.

Work out the perimeter of the rectangle. [5]

for $(x + 5)(x - 2) = 60$M1
for $x^2 + 3x - 70 = 0$A1
for a correct method to solve, e.g. $(x + 10)(x - 7) = 0$M1
$x = 7$ (may be implied by sides of 12 and 5; ignore $x = -10$ unless it is used for the perimeter)A1
$34$ (cm) (ft $2(2x + 3)$ with their positive $x$)B1
Answer: 34 cm
Examiner tip: x = −10 would give a negative side length, so only x = 7 can be used — then remember the question asks for the perimeter, not x.

4Grade 8–9HigherCalculator5 marks

Solve $\dfrac{3}{x + 1} + \dfrac{2}{x - 2} = 1$

Give your solutions correct to 3 significant figures. [5]

for multiplying through by $(x + 1)(x - 2)$, e.g. $3(x - 2) + 2(x + 1) = (x + 1)(x - 2)$M1
for expanding brackets correctly on at least one side, e.g. $5x - 4 = x^2 - x - 2$M1
for $x^2 - 6x + 2 = 0$A1
for correct use of the formula or completing the square, e.g. $x = 3 \pm \sqrt{7}$M1
$5.65$ and $0.354$A1
Answer: $x = 5.65$ or $x = 0.354$
Examiner tip: Multiply EVERY term by the common denominator — including the 1 on the right-hand side.

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