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GCSE / IGCSE Maths – Quadratics

Factorising, completing the square, the formula, graphs, simultaneous equations.

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12 questions  ·  67 marks  ·  answers included  ·  GCSE/IGCSE (Higher)
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This worksheet covers:

  • Factorising
  • Solving by factorising
  • Completing the square
  • The quadratic formula
  • Sketching quadratic graphs
  • Quadratic simultaneous equations
  • Problems leading to quadratics

Section A is non-calculator; a calculator may be used in Sections B and C. Allow about 60 minutes.

Questions

Section A – Skills

Non-calculator.
1

Factorise fully:(a) $x^2 + 7x + 12$(b) $x^2 – 5x – 14$(c) $2x^2 + 7x + 3$(d) $9x^2 – 25$

Show answer
(a) $(x+3)(x+4)$(b) $(x-7)(x+2)$(c) $(2x+1)(x+3)$(d) $(3x-5)(3x+5)$
[6]
2

Solve by factorising:(a) $x^2 – 3x – 10 = 0$(b) $3x^2 – 11x + 6 = 0$

Show answer
(a) $(x-5)(x+2) = 0$, $x = 5$ or $x = -2$(b) $(3x-2)(x-3) = 0$, $x = \frac{2}{3}$ or $x = 3$
[5]
3

(a) Write $x^2 + 6x + 5$ in the form $(x + a)^2 + b$.(b) Hence solve $x^2 + 6x + 5 = 0$.(c) Write $2x^2 – 8x – 5$ in the form $p(x + q)^2 + r$.

Show answer
(a) $(x+3)^2 – 4$(b) $(x+3)^2 = 4$, $x = -1$ or $x = -5$(c) $2(x-2)^2 – 13$
[6]
4

The graph of $y = x^2 – 8x + 5$ is drawn.(a) Write $x^2 – 8x + 5$ in completed-square form.(b) Write down the coordinates of the turning point of the graph.(c) Write down the equation of the line of symmetry.(d) Write down the $y$-intercept.

Show answer
(a) $(x-4)^2 – 11$(b) Minimum at $(4,\ -11)$(c) $x = 4$(d) $(0,\ 5)$
[5]

Section B – Exam-style

Calculator allowed. Give answers to 2 decimal places where appropriate.
5

Solve, giving your answers correct to 2 decimal places:(a) $2x^2 + 5x – 4 = 0$(b) $x^2 – 6x + 2 = 0$ (give exact answers in the form $a \pm \sqrt{b}$ as well).

Show answer
(a) $x = \dfrac{-5 \pm \sqrt{57}}{4}$: $x = 0.64$ or $x = -3.14$(b) $x = 3 \pm \sqrt{7}$: $x = 5.65$ or $x = 0.35$
[6]
6

Solve the simultaneous equations$y = x^2 – 3x + 1$ and $y = 2x – 3$.

Show answer
$x^2 – 5x + 4 = 0 \Rightarrow (x-1)(x-4) = 0$$x = 1,\ y = -1$ and $x = 4,\ y = 5$
[5]
7

Solve the simultaneous equations$x^2 + y^2 = 25$ and $y = x + 1$.

Show answer
$x^2 + (x+1)^2 = 25 \Rightarrow 2x^2 + 2x – 24 = 0 \Rightarrow (x+4)(x-3) = 0$$x = 3,\ y = 4$ and $x = -4,\ y = -3$
[6]
8

A rectangle has width $x$ cm and length $(x + 3)$ cm. Its area is $40$ cm$^2$.(a) Show that $x^2 + 3x – 40 = 0$.(b) Solve the equation and hence find the perimeter of the rectangle.

Show answer
(a) $x(x+3) = 40$(b) $(x+8)(x-5) = 0$; $x = 5$ (reject $-8$); perimeter $= 2(5 + 8) = 26$ cm
[5]
9

Solve $(x + 2)(x – 5) = 8$.

Show answer
$x^2 – 3x – 10 = 8 \Rightarrow x^2 – 3x – 18 = 0 \Rightarrow (x-6)(x+3) = 0$; $x = 6$ or $x = -3$
[4]
10

A ball is thrown upwards. Its height, $h$ metres, after $t$ seconds is $h = 20t – 5t^2$.(a) Find the times at which the ball is at a height of $15$ m.(b) Write $20t – 5t^2$ in the form $a – 5(t – b)^2$.(c) Hence write down the maximum height reached and the time at which it occurs.

Show answer
(a) $5t^2 – 20t + 15 = 0 \Rightarrow t^2 – 4t + 3 = 0$; $t = 1$ s and $t = 3$ s(b) $20 – 5(t-2)^2$(c) Maximum height $20$ m at $t = 2$ s
[6]

Section C – Challenge

Calculator allowed.
11

Solve $\dfrac{3}{x+1} + \dfrac{2}{x-2} = 1$, giving your answers correct to 3 significant figures.

Show answer
$3(x-2) + 2(x+1) = (x+1)(x-2) \Rightarrow x^2 – 6x + 2 = 0$$x = 3 \pm \sqrt{7}$: $x = 5.65$ or $x = 0.354$
[6]
12

A rectangular lawn measures $10$ m by $6$ m. It is surrounded on all sides by a path of uniform width $x$ metres. The total area of the lawn and path together is $96$ m$^2$.(a) Show that $x^2 + 8x – 9 = 0$.(b) Find the width of the path.(c) The equation $kx^2 + 4x + k = 0$ has equal roots. Find the possible values of $k$.

Show answer
(a) $(10 + 2x)(6 + 2x) = 96 \Rightarrow 4x^2 + 32x + 60 = 96 \Rightarrow 4x^2 + 32x – 36 = 0$, divide by $4$(b) $(x+9)(x-1) = 0$; $x = 1$ m (reject $-9$)(c) $b^2 – 4ac = 16 – 4k^2 = 0 \Rightarrow k = \pm 2$
[7]

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