AQA Level 2 Further Maths · Level 2 Certificate · Chapter 2: Algebra II · Lesson 4

Completing the square

About 75 minutesNo calculator62 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Expand and simplify $(x - 7)^{2} + 3$ .
[2 marks]
Answer
A2
Needed today
Expand and simplify $3 (x + 2)^{2} - 5$ .
[2 marks]
Answer
A3
Last lesson
Write $\frac{3}{x - 2} - \frac{2}{x + 1}$ as a single fraction.
[2 marks]
Answer
A4
From chapter 1
How many four-digit even numbers can be made from the digits $1$ to $9$ if digits may be repeated?
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

Write $x^{2} + 10 x + 7$ in the form $(x + a)^{2} + b$ .

Hint Halve the coefficient of

x

to get the number in the bracket.

(x + 5)^{2}

makes

x^{2} + 10 x + 25

, so take away the extra

25

B1
Write $x^{2} - 18 x + 50$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer

Example 2

Write $x^{2} + 3 x - 1$ in the form $(x + a)^{2} + b$ .

Hint Half of

3

\frac{3}{2}

, and

(\frac{3}{2})^{2} = \frac{9}{4}

. Keep the fractions; do not round.

B2
Write $x^{2} - 7 x + 5$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer

Example 3

Write $2 x^{2} - 12 x + 5$ in the form $a (x + b)^{2} + c$ .

Hint Take the

2

out of the

x^{2}

and

x

terms only. Complete the square inside the bracket, then multiply back out.

B3
Write $3 x^{2} + 24 x + 50$ in the form $a (x + b)^{2} + c$ .
[3 marks]
Answer

Example 4

Write $8 + 6 x - x^{2}$ in the form $p - (x - q)^{2}$ .

Hint Write it as

- (x^{2} - 6 x) + 8

. Complete the square inside, and watch the minus sign outside the bracket.

B4
Write $3 - 10 x - x^{2}$ in the form $p - (x + q)^{2}$ .
[2 marks]
Answer

Example 5

Work out the values of $a$ , $b$ and $c$ such that $4 x^{2} + b x + 1 \equiv a (x - 2)^{2} + c$ .

Hint Expand the right-hand side. Then make the coefficients of

x^{2}

, of

x

and the constants equal on both sides.

B5
Work out the values of $a$ , $b$ and $c$ such that $5 x^{2} + b x - 3 \equiv a (x + 3)^{2} + c$ .
[3 marks]
Answer

Practice

Level 2

C1
Write $x^{2} + 9 x + 20$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer
C2
Work out the values of $c$ and $d$ such that $x^{2} - c x + 11 \equiv (x - 3)^{2} + d$ .
[2 marks]
Answer

C3
Work out the values of $a$ , $b$ and $c$ such that $2 x^{2} + 5 x - 1 \equiv a (x + b)^{2} + c$ .
[3 marks]
Answer
C4
Work out the values of $p$ , $q$ and $r$ such that $9 - 8 x - 2 x^{2} \equiv p - q (x + r)^{2}$ .
[3 marks]
Answer
C5
By completing the square, find the minimum value of $x^{2} + 4 x + 9$ and the value of $x$ where it happens.
[3 marks]
Answer
C6
Use completing the square to make $x$ the subject of $y = x^{2} + 12 x$ , where $x > - 6$ .
[3 marks]
Answer

Exam-style questions

AQA exam style

D1
Ben writes " $2 x^{2} + 8 x + 3 = 2 (x + 2)^{2} - 1$ ".
Explain Ben's mistake. Write $2 x^{2} + 8 x + 3$ in the form $a (x + b)^{2} + c$ correctly.
[3 marks]
Answer

D2
This question is about $x^{2} + 8 x + 21$ .
(a)
Write $x^{2} + 8 x + 21$ in the form $(x + a)^{2} + b$ , where $a$ and $b$ are integers.
[2 marks]
Answer
(b)
Hence prove that $x^{2} + 8 x + 21$ is positive for all values of $x$ .
[2 marks]
(c)
Hence write down the greatest value of $\frac{10}{x ^{2} + 8 x + 21}$ .
[1 mark]
Answer
Total for D2: 5 marks
D3
Write $23 - 20 x - 4 x^{2}$ in the form $a - (b x + c)^{2}$ , where $a$ , $b$ and $c$ are positive integers.
[3 marks]
Answer
D4
$x^{2} + p x + q \equiv (x + 3)^{2} - 7$
(a)
Work out the values of $p$ and $q$ .
[2 marks]
Answer
(b)
Hence solve $x^{2} + p x + q = 0$ . Give your answers in surd form.
[2 marks]
Answer
Total for D4: 4 marks
D5
A rectangle has a perimeter of $20$ cm. Its width is $x$ cm.
Not drawn accurately
(a)
Show that the area, $A$ cm², of the rectangle is given by $A = 10 x - x^{2}$ .
[2 marks]
(b)
Write $10 x - x^{2}$ in the form $p - (x - q)^{2}$ .
[2 marks]
Answer
(c)
Hence write down the greatest possible area of the rectangle, and the width that gives it.
[2 marks]
Answer
Total for D5: 6 marks

Extension

Stretch

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

E1
Prove that $x^{2} + y^{2} - 4 x + 6 y + 14$ is positive for all values of $x$ and $y$ . Find its smallest value, and the values of $x$ and $y$ that give it.
[5 marks]
Hint 1 · What to try
Collect the $x$ terms together and the $y$ terms together. Complete the square on each group separately.
Hint 2 · The first line
$x^{2} - 4 x = (x - 2)^{2} - 4$ and $y^{2} + 6 y = (y + 3)^{2} - 9$ .
Hint 3 · The full method
So the expression is $(x - 2)^{2} + (y + 3)^{2} - 4 - 9 + 14 = (x - 2)^{2} + (y + 3)^{2} + 1$ . Both squares are at least $0$ , so it is at least $1$ , which is positive. The smallest value is $1$ , when $x = 2$ and $y = - 3$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a quadratic in the form $(x + a)^{2} + b$ , including with fractions.
I can complete the square when the coefficient of $x^{2}$ is not $1$ or is negative.
I can find unknown coefficients by comparing both sides of an identity.
I can use a completed square to find a minimum or maximum, or to prove a quadratic is positive.

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