GCSE Maths · Higher · Chapter 17: Quadratic equations

Completing the square

Grades 5–9About 70 minutesNo calculator85 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Expand $(x + 5)^{2}$ .
[1 mark]
Answer

A2
Expand $(x - 3)^{2}$ .
[1 mark]
Answer
A3
Solve $(x - 2)^{2} = 49$ . Give both answers.
[2 marks]
Answer
A4
Write $48$ in the form $a 3$ .
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Write $x^{2} + 16 x + 9$ in the form $(x + p)^{2} + q$ .

Hint Halve the coefficient of

x

to find

p

. Then take away the extra number the bracket makes.

B1
Write $x^{2} - 14 x + 30$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer

Example 2

Solve $x^{2} - 14 x + 37 = 0$ . Give your answers in the form $a \pm b 3$ .

Hint Complete the square first. Then take the square root of both sides, and remember a square root has two answers.

B2
Solve $x^{2} + 10 x + 13 = 0$ . Give your answers in the form $a \pm b 3$ .
[3 marks]
Answer

Example 3

Write $3 x^{2} + 12 x + 7$ in the form $a (x + b)^{2} + c$ .

Hint Take the 3 out of the first two terms only. Complete the square inside the bracket, then multiply back out.

B3
Write $2 x^{2} + 12 x + 5$ in the form $a (x + b)^{2} + c$ .
[3 marks]
Answer

Practice

Grades 6–7

Each question asks for something different. Show your working.

C1
Write $x^{2} + 18 x + 85$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer

C2
Write $x^{2} - 11 x + 20$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer
C3
Solve $x^{2} - 12 x + 31 = 0$ by completing the square. Give your answers in the form $a \pm b$ .
[3 marks]
Answer
C4
Find the coordinates of the turning point of the curve $y = x^{2} - 10 x + 31$ .
[2 marks]
Answer
C5
Write $3 x^{2} - 18 x + 20$ in the form $a (x + b)^{2} + c$ .
[3 marks]
Answer
C6
Show that $x^{2} - 10 x + 34$ is positive for every value of $x$ .
[2 marks]
C7
$y = x^{2} - 12 x + 11$
(a)
Write $x^{2} - 12 x + 11$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer
(b)
Sketch the curve on the axes. Mark the coordinates of the turning point and of the points where the curve meets the axes.
[2 marks]
Total for C7: 4 marks

Exam-style questions

Grades 7–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$x^{2} + 12 x + k \equiv (x + a)^{2} - 3$ for all values of $x$ .
Find the value of $a$ and the value of $k$ .
[3 marks]
Answer $a = \dots\dots$ $k = \dots\dots$

D2
The diagram shows a sketch of the curve $y = x^{2} + b x + c$ .
The turning point of the curve is $(3, - 7)$ .
Find the value of $b$ and the value of $c$ .
[3 marks]
Not drawn accurately
Answer $b = \dots\dots$ $c = \dots\dots$
D3
Sam says, "The minimum value of $x^{2} - 14 x + 20$ is 20, because the smallest $x^{2}$ can be is 0."
Sam is wrong. Explain why, and find the minimum value of $x^{2} - 14 x + 20$ .
[3 marks]
Answer
D4
Show that the solutions of $x^{2} + 12 x + 29 = 0$ are $x = - 6 \pm 7$ .
[3 marks]
D5
Write $2 x^{2} - 16 x + 35$ in the form $a (x + b)^{2} + c$ .
Hence write down the coordinates of the turning point of the curve $y = 2 x^{2} - 16 x + 35$ .
[4 marks]
Answer $\dots\dots\dots\dots$ turning point $(\dots\dots, \dots\dots)$
D6
The diagram shows a sketch of the curve $y = x^{2} + p x + q$ .
The curve meets the $y$ -axis at $(0, 13)$ . The line $x = 5$ goes through the turning point.
Find the coordinates of the turning point.
[4 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$
D7
The diagram shows a rectangle. All lengths are in centimetres.
The area of the rectangle is 40 cm².
Not drawn accurately
(a)
Show that $x^{2} + 6 x - 47 = 0$ .
[2 marks]
(b)
By completing the square, find the value of $x$ . Give your answer in the form $a + b 14$ . You must explain why there is only one answer.
[4 marks]
Answer $x = \dots\dots\dots\dots$
Total for D7: 6 marks
D8
$f (x) = x^{2} - 6 x + 17$
(a)
Write $f (x)$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer
(b)
Hence explain why the equation $f (x) = 5$ has no real solutions.
[2 marks]
Total for D8: 4 marks
D9
The curve $y = x^{2} + 10 x + 22$ is translated by the vector $(23)$ .
Show that the new curve touches the $x$ -axis, and find the point where it touches.
[4 marks]
Answer $(\dots\dots, \dots\dots)$
D10
A ball is thrown upwards. Its height, $h$ metres, after $t$ seconds is given by
$h = 30 t - 5 t^{2}$
(a)
Write $h$ in the form $c - 5 (t - d)^{2}$ .
[2 marks]
Answer
(b)
Write down the greatest height of the ball, and the time at which it is reached.
[2 marks]
Answer $\dots\dots$ m after $\dots\dots$ s
(c)
For how many seconds is the ball more than 25 m above the ground?
[2 marks]
Answer
Total for D10: 6 marks

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Find the largest value that $\frac{1}{x ^{2} - 10 x + 29}$ can take, and the value of $x$ that gives it.
[3 marks]
Hint 1 · What to try
A fraction with 1 on top is largest when its bottom is smallest.
Hint 2 · The first line
$x^{2} - 10 x + 29 = (x - 5)^{2} + 4$
Hint 3 · The full method
The bottom is never less than 4, and it equals 4 when $x = 5$ . So the largest value is $\frac{1}{4}$ .
Answer

E2
Find the values of $k$ for which the equation $x^{2} + 6 x + k = 0$ has no real solutions.
[3 marks]
Hint 1 · What to try
Complete the square, then move the number to the other side.
Hint 2 · The first line
$(x + 3)^{2} = 9 - k$
Hint 3 · The full method
A square is never negative, so there is no solution when $9 - k < 0$ , which is when $k > 9$ .
Answer
E3
The curve $y = x^{2} - 2 a x + 3 a^{2} + 4$ , where $a$ is a constant.
Find the coordinates of the turning point in terms of $a$ , and show that the curve never meets the $x$ -axis.
[4 marks]
Hint 1 · What to try
Treat $a$ as a number. Halve the coefficient of $x$ in the usual way.
Hint 2 · The first line
$y = (x - a)^{2} - a^{2} + 3 a^{2} + 4$
Hint 3 · The full method
$y = (x - a)^{2} + 2 a^{2} + 4$ , so the turning point is $(a, 2 a^{2} + 4)$ . Since $2 a^{2} + 4$ is at least 4, $y$ is always at least 4.
E4
The numbers $x$ and $y$ satisfy $x^{2} + y^{2} - 4 x + 8 y + 20 = 0$ .
Find the value of $x y$ .
[4 marks]
Hint 1 · What to try
Complete the square twice: once for the $x$ terms and once for the $y$ terms.
Hint 2 · The first line
$(x - 2)^{2} + (y + 4)^{2} = 0$
Hint 3 · The full method
Two squares that add to 0 must both be 0. So $x = 2$ and $y = - 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a quadratic in the form $(x + p)^{2} + q$ , including when the coefficient of $x^{2}$ is not 1.
I can solve a quadratic exactly by completing the square.
I can find the turning point and the minimum value from completed-square form.

Answers and mark scheme

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