Cambridge International Mathematics

Try as much as we like, we will not be able to solve quadratic

equations such asx^2 +4x¡7=0using the factorisation methods

already practised. This is because the solutions are not rationals.

Not all

quadratics have

simple factors.

Consequently, thequadratic formulahas been developed:

If ax^2 +bx+c=0 where a 6 =0, then x=

¡b§

p

b^2 ¡ 4 ac

2 a

.

Proof: If ax^2 +bx+c=0

then x^2 +

b

a

x+

c

a

=0 fdividing each term bya,asa 6 =0g

) x^2 +

b

a

x =¡

c

a

) x^2 +

b

a

x+

μ

b

2 a

¶ 2

=¡

c

a

+

μ

b

2 a

¶ 2

fcompleting the square on LHSg

)

μ

x+

b

2 a

¶ 2

=¡

c

a

μ

4 a

4 a

¶

+

b^2

4 a^2

)

μ

x+

b

2 a

¶ 2

=

b^2 ¡ 4 ac

4 a^2

) x+

b

2 a

=§

r

b^2 ¡ 4 ac

4 a^2

) x=¡

b

2 a

§

p

b^2 ¡ 4 ac

2 a

) x=

¡b§

p

b^2 ¡ 4 ac

2 a

To demonstrate the validity of this formula, consider the equation x^2 ¡ 3 x+2=0.

By factorisation: x^2 ¡ 3 x+2=0

) (x¡1)(x¡2) = 0

) x=1or 2

By formula: a=1,b=¡ 3 ,c=2

) x=

¡(¡3)§

p

(¡3)^2 ¡4(1)(2)

2

) x=

3 §

p

9 ¡ 8

2

) x=

3 § 1

2

) x=2or 1

We can see that factorisation is quicker if the quadratic can indeed be factorised, but the quadratic formula

provides an alternative for when it cannot be factorised.

C THE QUADRATIC FORMULA [2.10]

Quadratic equations and functions (Chapter 21) 427

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Y:\HAESE\IGCSE01\IG01_21\427IGCSE01_21.CDR Monday, 27 October 2008 2:09:07 PM PETER