GCSE Maths · Higher · Chapter 17: Quadratic equations · Lesson 4

Completing the square

Grades 5–8About 45 minutesCalculator allowed36 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Expand $(x - 6)^{2}$ .
[1 mark]
Answer
A2
From chapter 12
Two similar solids have volumes in the ratio $1 : 64$ . Find the ratio of their lengths.
[1 mark]
Answer
A3
Last lesson
Work out $b^{2} - 4 a c$ for $x^{2} + x + 4 = 0$ .
[1 mark]
Answer
A4
From chapter 13
A spinner lands on red 18 times in 100 spins. Find the relative frequency of red.
[1 mark]
Answer

Grades 5–6

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

Example 1

Write $x^{2} + 20 x + 7$ in the form $(x + a)^{2} + b$ .

Hint Halve the

20

to get the number inside the bracket. Then take off

1 0^{2}

B1
Write $x^{2} - 16 x + 55$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $(x + \dots\dots)^{2} + \dots\dots$

Example 2

Solve $x^{2} + 14 x + 11 = 0$ by completing the square. Give your answers in the form $a \pm b$ .

Hint Write the left side as

(x + 7)^{2} - 38

, then make the bracket the subject.

B2
Solve $x^{2} - 18 x + 4 = 0$ by completing the square. Give your answers in the form $a \pm b$ .
[3 marks]
Answer $x = \dots\dots \pm \dots\dots$

Grades 6–7

C1
Show that $x^{2} - 4 x + 11 > 0$ for every value of $x$ .
[2 marks]
C2
Write $x^{2} - 7 x + 3$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $(x + \dots\dots)^{2} + \dots\dots$

C3
Write $3 x^{2} + 24 x + 5$ in the form $a (x + b)^{2} + c$ .
[3 marks]
Answer $\dots (x + \dots)^{2} + \dots$
C4
Work out the turning point of $y = x^{2} - 22 x + 130$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
C5
Given that $2 x^{2} + 20 x + q \equiv a (x + b)^{2} - 3$ , find the values of $a$ , $b$ and $q$ .
[3 marks]
Answer $a = \dots\dots b = \dots\dots q = \dots\dots$
C6
The sketch shows the curve $y = (x - 7)^{2} - 16$ . It crosses the $x$ -axis at $A$ and $B$ , and the $y$ -axis at $C$ . Work out the coordinates of $A$ , $B$ and $C$ .
[3 marks]
Not drawn accurately
Answer $A = (\dots, \dots) B = (\dots, \dots) C = (\dots, \dots)$

Grades 6–8

D1
Dev writes $x^{2} - 10 x + 3 = (x - 5)^{2} + 3$ .
Explain Dev's mistake, and write $x^{2} - 10 x + 3$ correctly in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $(x + \dots\dots)^{2} + \dots\dots$

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a quadratic in the form $(x + a)^{2} + b$ .
I can solve a quadratic exactly by completing the square.
I can read the turning point from the completed-square form.

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