Number and Algebra 317

Example 10 Factorising non-monic quadratics

Factorise:a 6x2 + 19x + 10 b 9x2 + 6x − 8

SOLUTION EXPLANATION

a 6x2 + 19x + 10 = 6x2 + 15x + 4x + 10 = 3x(2x + 5) + 2(2x + 5) = (2x + 5)(3x + 2)

a × c = 6 × 10 = 60, choose 15 and 4.15 × 4 = 60 and 15 + 4 = 19.Factorise by grouping.

b 9x2 + 6x − 8 = 9x2 + 12x − 6x − 8 = 3x(3x + 4) − 2(3x + 4) = (3x + 4)(3x − 2)

a × c = 9 × (-8) = -72, choose 12 and -6.12 × (-6) = -72 and 12 + (-6) = 6.

Example 11 Simplifying algebraic fractions

Simplify 4 9

10 13 325 10 110 17 3

2

2

2

2x

x xx xx x

−+ −

× − +− +

.

SOLUTION EXPLANATION

4 910 13 3

25 10 110 17 3

2 3

2

2

2

2x

x xx xx x

x

−+ −

× − +− +

=+( ) 11 1

1 1

1 12 3

2 3 5 1

5 1 5 1

2 3

( )

( ) ( )

( ) ( )

(

x

x x

x x

x

−+ −

×− −− )) ( )1 15 1

1

x −=

First factorise all quadratics.Cancel to simplify.

Exercise 5D1 Complete this table.

ax 2 + bx + c a × c Two numbers that multiply to give a × c and add to give b

6x 2 + 13x + 6 36 9 and ____

8x 2 + 18x + 4 32

12x 2 + x − 6 -8 and ____

10x 2 − 11x − 6

21x 2 − 20x + 4 -6 and ____

15x 2 − 13x + 2

2 Factorise by grouping pairs.a x2 + 2x + 5x + 10 b x2 + 4x + 6x + 24 c x2 + 3x + 7x + 21d x2 − 7x − 2x + 14 e x2 − 3x − 4x + 12 f x2 − 5x + 3x − 15g 6x2 − 8x + 3x − 4 h 3x2 − 12x + 2x − 8 i 8x2 − 4x + 6x − 3j 5x2 + 20x − 2x − 8 k 10x2 + 12x − 15x − 18 l 12x2 − 6x − 10x + 5

Unde

rsta

ndin

g

ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.

© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press

Chapter 5 Quadratic equations318

Flue

ncy

3 Factorise the following.a 3x2 + 10x + 3 b 2x2 + 3x + 1 c 3x2 + 8x + 4d 3x2 − 5x + 2 e 2x2 − 11x + 5 f 5x2 + 2x − 3g 3x2 − 11x − 4 h 3x2 − 2x − 1 i 7x2 + 2x − 5j 2x2 − 9x + 7 k 3x2 + 2x − 8 l 2x2 + 5x − 12m 2x2 − 9x − 5 n 13x2 − 7x − 6 o 5x2 − 22x + 8p 8x2 − 14x + 5 q 6x2 + x − 12 r 10x2 + 11x − 6s 6x2 + 13x + 6 t 4x2 − 5x + 1 u 8x2 − 14x + 5v 8x2 − 26x + 15 w 6x2 − 13x + 6 x 9x2 + 9x − 10

4 Factorise the following.a 18x2 + 27x + 10 b 20x2 + 39x + 18 c 21x2 + 22x − 8d 30x2 + 13x − 10 e 40x2 − x − 6 f 28x2 − 13x − 6g 24x2 − 38x + 15 h 45x2 − 46x + 8 i 25x2 − 50x + 16

Example 10

Prob

lem

-sol

ving

5 Factorise by first taking out the common factor.a 6x2 + 38x + 40 b 6x2 − 15x − 36 c 48x2 − 18x − 3d 32x2 − 88x + 60 e 16x2 − 24x + 8 f 90x2 + 90x − 100g -50x2 − 115x − 60 h 12x2 − 36x + 27 i 20x2 − 25x + 5

6 Simplify by first factorising.

a 6 353 7

2x xx− −

+ b 8 10 3

2 3x x

x+ −

+ c 9 21 10

3 5

2x xx

− +−

d 10 2

15 7 22x

x x−

+ − e

4 6

14 17 62

x

x x

++ −

f 20 12

10 21 92

x

x x

−− +

g 2 11 12

6 11 3

2

2

x x

x x

+ ++ +

h 12 1

8 14 3

2

2

x x

x x

− −+ +

i 10 3 4

14 11 2

2

2

x x

x x

+ −− +

j 9 4

15 4 4

2

2

x

x x

−+ −

k 14 19 3

49 1

2

2

x x

x

+ −−

l 8 2 15

16 25

2

2

x x

x

− −−

7 A cable is suspended across a farm channel. The height (h) in metres of the cable above the water surface is modelled by the equation h = 3x2 − 21x + 30 where x metres is the distance from one side of the channel.a Factorise the right-hand side of the

equation.b Determine the height of the cable when

x = 3. Interpret this result.c Determine where the cable is at the

level of the water surface.

ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.

© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press