KS3 Maths · Year 8 · Chapter 8: Algebra · Lesson 5

The difference of two squares

Year 8About 70 minutesNo calculator47 marks

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Date

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Quick questions to start.

A1
Last lesson
Factorise $x^{2} - 7 x - 78$ .
[1 mark]
Answer
A2
Needed today
Work out $1 7^{2}$ .
[1 mark]
Answer
A3
Lesson 1
Expand and simplify $(3 x + 2) (3 x - 2)$ .
[1 mark]
Answer
A4
Earlier
Factorise $6 x^{2} - 9 x$ fully.
[1 mark]
Answer

Example, then your turn

Year 8

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

Example 1

Factorise $x^{2} - 289$ .

Hint There is no

x

term, so you want two numbers that add to

0

and multiply to

- 289

289

is a square number.

B1
Factorise $x^{2} - 441$ .
[2 marks]
Answer

Example 2

Factorise $49 a^{2} - b^{2}$ .

Hint

a^{2} - b^{2} = (a + b) (a - b)

works for any two squares. Write

49 a^{2}

as a square.

B2
Factorise $p^{2} - 36 q^{2}$ .
[2 marks]
Answer

Example 3

Factorise $4 x^{2} - 81$ .

Hint Both terms are squares:

4 x^{2} = (2 x)^{2}

and

81 = 9^{2}

B3
Factorise $25 x^{2} - 36$ .
[2 marks]
Answer

Example 4

Factorise fully $3 x^{2} - 147$ .

Hint

147

is not a square number. Take out the common factor

3

first.

B4
Factorise fully $2 x^{2} - 288$ .
[3 marks]
Answer

Example 5

Work out $4 7^{2} - 4 3^{2}$ without a calculator.

Hint

a^{2} - b^{2} = (a + b) (a - b)

. Use

a = 47

and

b = 43

B5
Work out $8. 6^{2} - 1. 4^{2}$ without a calculator.
[2 marks]
Answer

Practice

Year 8

C1
Factorise $y^{2} - 324$ .
[2 marks]
Answer
C2
Fill in the missing numbers: $□ x^{2} - 121 = (2 x + □) (2 x - 11)$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

C3
A square of side $17$ cm is cut from a square of side $23$ cm. Work out the area of the shaded part without a calculator.
[2 marks]
Not drawn accurately
Answer $\dots\dots$ cm²
C4
Two of these can be factorised as the difference of two squares. Which two? Factorise them.
A: $x^{2} + 400$
B: $9 x^{2} - 1$
C: $x^{2} - 30 x$
D: $64 - x^{2}$
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
C5
Simplify $\frac{x ^{2} - 576}{x - 24}$ .
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Ana says: " $36 x^{2} - 25 = (36 x + 25) (36 x - 25)$ ."
Ana is wrong. Explain why, and factorise $36 x^{2} - 25$ correctly.
[3 marks]
Answer $36 x^{2} - 25 = \dots\dots$

D2
A square of side $7$ cm is cut from a square of side $2 x$ cm.
Not drawn accurately
(a)
Show that the area of the shaded part is $(2 x + 7) (2 x - 7)$ cm².
[2 marks]
Answer
(b)
Use your answer to part (a) to work out the shaded area when $x = 10$ .
[2 marks]
Answer $\dots\dots$ cm²
Total for D2: 4 marks
D3
$n$ is a whole number. Show that $(n + 5)^{2} - (n - 5)^{2}$ is always a multiple of $20$ .
[4 marks]
Answer
D4
The rectangle is $(x + 19)$ cm long and $(x - 19)$ cm wide. A square has sides of $x$ cm.
Not drawn accurately
(a)
Show that the area of the square is $361$ cm² more than the area of the rectangle.
[2 marks]
Answer
(b)
The area of the rectangle is $264$ cm². Work out the value of $x$ .
[3 marks]
Answer
Total for D4: 5 marks
D5
Factorise fully $48 x^{2} - 75$ .
[3 marks]
Answer

Extension

Stretch

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

E1
$a$ and $b$ are whole numbers with $a > b > 0$ , and $a^{2} - b^{2} = 91$ . Find both possible pairs of values of $a$ and $b$ .
[3 marks]
Hint 1 · What to try
Factorise $a^{2} - b^{2}$ .
Hint 2 · The first line
$(a + b) (a - b) = 91$ . Write $91$ as a product of two whole numbers in every way you can.
Hint 3 · The full method
For each way, $a + b$ is the bigger number and $a - b$ the smaller. Solve for $a$ and $b$ .
Answer $a = \dots$ , $b = \dots$ and $a = \dots$ , $b = \dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can factorise $x^{2} - a^{2}$ as $(x + a) (x - a)$ .
I can spot a square such as $25 x^{2} = (5 x)^{2}$ inside a difference of two squares.
I can take out a common factor first and then factorise fully.
I can use the difference of two squares to work out areas and calculations.

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