KS3 Maths · Year 8 · Chapter 8: Algebra · Lesson 4

Factorising quadratics with negative coefficients

Year 8About 75 minutesNo calculator47 marks

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Date

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Quick questions to start.

A1
Last lesson
Factorise $x^{2} + 23 x + 120$ .
[1 mark]
Answer
A2
Needed today
Find two numbers that multiply to $- 24$ and add to $5$ .
[1 mark]
Answer
A3
Lesson 1
Expand and simplify $(x - 3) (x + 13)$ .
[1 mark]
Answer
A4
Year 7
Work out the value of $x^{2} - 3 x$ when $x = - 6$ .
[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 $x^{2} + 4 x - 96$ .

Hint The product

- 96

is negative, so one number is positive and one is negative. The sum

+ 4

is positive, so the positive number is the bigger one.

B1
Factorise $x^{2} + x - 90$ .
[2 marks]
Answer

Example 2

Factorise $x^{2} - 19 x + 84$ .

Hint The product

+ 84

is positive, so the two numbers have the same sign. The sum

- 19

is negative, so both are negative.

B2
Factorise $x^{2} - 15 x + 26$ .
[2 marks]
Answer

Example 3

Factorise $x^{2} - 10 x - 56$ .

Hint The product is negative, so one number of each sign. The sum is negative, so the negative number is the bigger one.

B3
Factorise $x^{2} - 6 x - 55$ .
[2 marks]
Answer

Example 4

Factorise $x^{2} - 26 x + 169$ .

Hint Look for two numbers with product

169

and sum

- 26

. They may be the same number.

B4
Factorise $x^{2} - 36 x + 324$ .
[2 marks]
Answer

Example 5

Factorise fully $135 - 42 x + 3 x^{2}$ .

Hint Write the terms in the usual order,

3 x^{2} - 42 x + 135

. Every term is a multiple of

3

, so take out the

3

first.

B5
Factorise fully $4 x^{2} - 20 x - 200$ .
[3 marks]
Answer

Practice

Year 8

C1
Factorise $x^{2} + 6 x - 72$ .
[2 marks]
Answer
C2
Fill in the missing numbers: $x^{2} - □ x - 60 = (x + 4) (x - □)$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

C3
The diagram shows a square of side $x$ . A strip of width $a$ is cut from the top and a strip of width $b$ from the side. The shaded rectangle that is left has area $x^{2} - 17 x + 72$ . Work out $a$ and $b$ , where $a < b$ .
[2 marks]
Not drawn accurately
Answer $a = \dots\dots$ $b = \dots\dots$
C4
Factorise $x^{2} - 11 x - 12$ .
[2 marks]
Answer
C5
Factorise $- 52 + 9 x + x^{2}$ .
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Mia says: " $x^{2} - x - 132 = (x - 11) (x + 12)$ , because $- 11 \times 12 = - 132$ ."
Mia is wrong. Explain why, and factorise $x^{2} - x - 132$ correctly.
[3 marks]
Answer $x^{2} - x - 132 = \dots\dots$

D2
The area of this rectangle is $(x^{2} - 2 x - 120)$ cm². Its length is $(x + 10)$ cm.
Not drawn accurately
(a)
Write down an expression for the width of the rectangle.
[2 marks]
Answer $\dots\dots$ cm
(b)
The width of the rectangle is $3$ cm. Work out the area of the rectangle.
[3 marks]
Answer $\dots\dots$ cm²
Total for D2: 5 marks
D3
Show that $x^{2} - 32 x + 256$ can never be negative.
[3 marks]
Answer
D4
$n$ is a whole number. Show that $n^{2} - 25 n + 156$ is always even.
[4 marks]
Answer
D5
Here is an expression: $5 x^{2} - 65 x + 110$ .
(a)
Factorise $5 x^{2} - 65 x + 110$ fully.
[3 marks]
Answer
(b)
Use your answer to work out the value of $5 x^{2} - 65 x + 110$ when $x = 12$ .
[1 mark]
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 $(x + 3)^{2} - 2 (x + 3) - 80$ . Write each bracket as simply as you can.
[3 marks]
Hint 1 · What to try
Write $y$ in place of $(x + 3)$ .
Hint 2 · The first line
Now factorise $y^{2} - 2 y - 80$ : find two numbers with product $- 80$ and sum $- 2$ .
Hint 3 · The full method
Put $x + 3$ back in place of $y$ , and simplify each bracket.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the signs of the product and the sum to choose the two numbers.
I can factorise $x^{2} + b x + c$ when $b$ or $c$ is negative.
I can rearrange the terms and take out a common factor before I factorise.
I can use factorising to show an expression is never negative or always even.

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