Cambridge International Mathematics

Equations of the form ax+b=0where a 6 =0are calledlinear equationsand haveonly onesolution.

For example, 3 x¡2=0is the linear equation witha=3andb=¡ 2 : It has the solutionx=^23.

Equations of the form ax^2 +bx+c=0wherea 6 =0are calledquadratic equations.

They may havetwo,oneorzerosolutions.

Here are some simple quadratic equations which clearly show the truth of this statement:

Equation ax^2 +bx+c=0form a b c Solutions

x^2 ¡4=0 x^2 +0x¡4=0 1 0 ¡ 4 x=2orx=¡ 2 two

(x¡2)^2 =0 x^2 ¡ 4 x+4=0 1 ¡ 4 4 x=2 one

x^2 +4=0 x^2 +0x+4=0 1 0 4 none asx^2 is always> 0 zero

Now consider the example x^2 +3x¡10 = 0:

If x=2, x^2 +3x¡ 10

=2^2 +3£ 2 ¡ 10

=4+6¡ 10

=0

and if x=¡ 5 , x^2 +3x¡ 10

=(¡5)^2 +3£(¡5)¡ 10

=25¡ 15 ¡ 10

=0

x=2andx=¡ 5 both satisfy the equation x^2 +3x¡10 = 0, so we say they are bothsolutions.

We will discuss several methods for solving quadratic equations, and apply them to practical problems.

EQUATIONS OF THE FORM x^2 =k

x^2 =k is the simplest form of a quadratic equation.

Consider the equation x^2 =7:

Now

p

7 £

p

7=7,sox=

p

7 is one solution,

and (¡

p

7)£(¡

p

7) = 7,sox=¡

p

7 is also a solution.

Thus, if x^2 =7, then x=§

p

7 :

If x^2 =k then

8

><

>:

x=§

p

k ifk> 0

x=0 ifk=0

there areno real solutions ifk< 0 :

Example 1 Self Tutor

Solve forx: a 2 x^2 +1=15 b 2 ¡ 3 x^2 =8

a 2 x^2 +1=15

) 2 x^2 =14 fsubtracting 1 from both sidesg

) x^2 =7 fdividing both sides by 2 g

) x=§

p

7

A QUADRATIC EQUATIONS [2.10]

§

p

7 is read as

‘plus or minus the

square root of ’ 7

422 Quadratic equations and functions (Chapter 21)

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Y:\HAESE\IGCSE01\IG01_21\422IGCSE01_21.CDR Monday, 27 October 2008 2:08:52 PM PETER