KS3 Maths · Year 8 · Chapter 8: Algebra · Lesson 3

Factorising quadratics with positive coefficients

Year 8About 70 minutesNo calculator49 marks

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Date

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Quick questions to start.

A1
Last lesson
Expand and simplify $x (x + 1) (x + 2)$ .
[1 mark]
Answer
A2
Needed today
Write down every pair of whole numbers that multiply to make $48$ .
[1 mark]
Answer
A3
Needed today
Find two numbers that add to $15$ and multiply to $56$ .
[1 mark]
Answer
A4
Year 7
Factorise $6 x + 15$ .
[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 $x^{2} + 19 x + 90$ .

Hint

(x + a) (x + b) = x^{2} + (a + b) x + ab

. Find two numbers with product

90

and sum

19

B1
Factorise $x^{2} + 18 x + 77$ .
[2 marks]
Answer

Example 2

Factorise $x^{2} + 19 x + 48$ .

Hint List the pairs that multiply to

48

in order:

1

and

48

2

and

24

3

and

16

4

and

12

6

and

8

. Add each pair.

B2
Factorise $x^{2} + 15 x + 36$ .
[2 marks]
Answer

Example 3

Factorise $x^{2} + 30 x + 225$ .

Hint Look for two numbers with product

225

and sum

30

. They may be the same number.

B3
Factorise $x^{2} + 40 x + 400$ .
[2 marks]
Answer

Example 4

Factorise fully $4 x^{2} + 48 x + 44$ .

Hint Every term is a multiple of

4

. Take out the

4

first, then factorise what is left in the bracket.

B4
Factorise fully $2 x^{2} + 46 x + 84$ .
[3 marks]
Answer

Example 5

The area of this rectangle is $x^{2} + 21 x + 110$ . Its sides are $(x + a)$ and $(x + b)$ , where $a < b$ . Work out $a$ and $b$ .

Not drawn accurately

Hint Factorise the area. The two brackets are the two sides.

B5
The area of this right-angled triangle is $\frac{1}{2} (x^{2} + 20 x + 96)$ . Its two shorter sides are $(x + a)$ and $(x + b)$ , where $a < b$ . Work out $a$ and $b$ .
[3 marks]
Not drawn accurately
Answer $a = \dots\dots$ $b = \dots\dots$

Practice

Year 8

C1
Factorise $x^{2} + 16 x + 55$ .
[2 marks]
Answer
C2
Fill in the missing numbers: $x^{2} + □ x + 72 = (x + 6) (x + □)$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

C3
Only one of these cannot be factorised using whole numbers. Which one? Give a reason.
A: $x^{2} + 19 x + 78$
B: $x^{2} + 29 x + 78$
C: $x^{2} + 21 x + 78$
[2 marks]
Answer $\dots\dots$ because $\dots\dots$
C4
Factorise $x^{2} + 19 x + 88$ .
[2 marks]
Answer
C5
Factorise fully $3 x^{2} + 57 x + 180$ .
[3 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Leo says: " $x^{2} + 17 x + 60 = (x + 3) (x + 20)$ , because $3 \times 20 = 60$ ."
Leo is wrong. Explain why, and factorise $x^{2} + 17 x + 60$ correctly.
[3 marks]
Answer $x^{2} + 17 x + 60 = \dots\dots$

D2
The area of this rectangle is $(x^{2} + 23 x + 132)$ cm². Its length is $(x + 12)$ cm.
Not drawn accurately
(a)
Write down an expression for the width of the rectangle.
[2 marks]
Answer $\dots\dots$ cm
(b)
The perimeter of the rectangle is $86$ cm. Work out the value of $x$ .
[3 marks]
Answer
Total for D2: 5 marks
D3
$x^{2} + k x + 40$ can be factorised as $(x + a) (x + b)$ , where $a$ , $b$ and $k$ are positive whole numbers. Write down every possible value of $k$ .
[4 marks]
Answer
D4
Simplify $\frac{x ^{2} + 20 x + 99}{x + 9}$ .
[3 marks]
Answer
D5
Here is an expression: $x^{2} + 21 x + 108$ .
(a)
Factorise $x^{2} + 21 x + 108$ .
[2 marks]
Answer
(b)
Use your answer with $x = 100$ to work out $109 \times 112$ . Show your working.
[2 marks]
Answer
Total for D5: 4 marks

Extension

Stretch

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

E1
Factorise $x^{4} + 18 x^{2} + 80$ .
[3 marks]
Hint 1 · What to try
$x^{4} = (x^{2})^{2}$ . Write $y$ in place of $x^{2}$ .
Hint 2 · The first line
Now factorise $y^{2} + 18 y + 80$ : find two numbers with product $80$ and sum $18$ .
Hint 3 · The full method
Put $x^{2}$ back in place of $y$ in each bracket.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can factorise $x^{2} + b x + c$ by finding two numbers with product $c$ and sum $b$ .
I can spot a perfect square and write it as $(x + a)^{2}$ .
I can take out a common factor first and then factorise.
I can use factorising to find sides, simplify fractions and do arithmetic.

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