Cambridge International Mathematics

USE OF THE QUADRATIC FORMULA

If b^2 ¡ 4 ac is a rational perfect square then

p

b^2 ¡ 4 ac will be rational, and so the solutions of the

quadratic will also be rational. In such instances, it is preferable to solve the quadratic by factorisation.

For example, 6 x^2 ¡ 13 x¡8=0has b^2 ¡ 4 ac= 169¡4(6)(¡8) = 361 = 19^2 , so we should solve this

equation by factorising 6 x^2 ¡ 13 x¡ 8 into (3x¡8) (2x+1).

Example 7 Self Tutor

Solve forx:

a x^2 ¡ 2 x¡2=0 b 2 x^2 +3x¡4=0

a x^2 ¡ 2 x¡2=0 has

a=1, b=¡ 2 , c=¡ 2

) x=

¡(¡2)§

p

(¡2)^2 ¡4(1)(¡2)

2(1)

) x=

2 §

p

4+8

2

) x=

2 §

p

12

2

) x=

2 § 2

p

3

2

) x=1§

p

3 fexact formg

b 2 x^2 +3x¡4=0 has

a=2, b=3, c=¡ 4

) x=

¡ 3 §

p

32 ¡4(2)(¡4)

2(2)

) x=

¡ 3 §

p

9+32

4

) x=

¡ 3 §

p

41

4

So, x¼ 0 : 85 or ¼¡ 2 : 35

fcorrect to 2 decimal placesg

EXERCISE 21C.1

1 Use the quadratic formula to solve forx, giving exact answers::

a x^2 +4x¡3=0 b x^2 +6x+1=0 c x^2 +4x¡7=0

d x^2 +2x=2 e x^2 +2=6x f x^2 =4x+1

g x^2 +1=3x h x^2 +8x+5=0 i 2 x^2 =2x+1

j 9 x^2 =6x+1 k 25 x^2 +1=20x l 2 x^2 +6x+1=0

2 Use the quadratic formula to solve forx, giving answers correct to 2 decimal places:

a x^2 ¡ 6 x+4=0 b 2 x^2 +4x¡1=0 c 5 x^2 +2x¡4=0

d 3 x^2 +2x¡2=0 e x+

1

x

=3 f x¡

3

x

=1

3 Use the quadratic formula to solve forx:

a (x+ 2)(x¡1) = 5 b (x+1)^2 =3¡x^2 c

x+1

x

=

x

2

d x+

1

x+2

=4 e 3 x¡

4

x+1

=10 f

x+2

x¡ 1

=

3 x

x+1

428 Quadratic equations and functions (Chapter 21)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_21\428IGCSE01_21.CDR Monday, 27 October 2008 2:09:10 PM PETER