Cambridge Additional Mathematics

68 Quadratics (Chapter 3)

An alternative way to solve equations like x^2 +4x+1=0 is by ‘completing the square’.

Equations of the form ax^2 +bx+c=0can be converted to the form (x+p)^2 =q, from which the

solutions are easy to obtain.

Example 4 Self Tutor

Solve exactly forx:

a (x+2)^2 =7 b (x¡1)^2 =¡ 5

a (x+2)^2 =7

) x+2=§

p

7

) x=¡ 2 §

p

7

b (x¡1)^2 =¡ 5

has no real solutions since

the square (x¡1)^2 cannot

be negative.

The completed square form of an equation is (x+p)^2 =q.

If we expand this out, x^2 +2px+p^2 =q.

Notice that thecoefficient ofxequals 2 p. Therefore,pis half the coefficient ofxin the expanded form.

If we have x^2 +2px=q, then we “complete the square” by adding inp^2 to both sides of the equation.

Example 5 Self Tutor

Solve for exact values ofx: x^2 +4x+1=0

x^2 +4x+1=0

) x^2 +4x=¡ 1 fput the constant on the RHSg

) x^2 +4x+2^2 =¡ 1 +2^2 fcompleting the squareg

) (x+2)^2 =3 ffactorising LHSg

) x+2=§

p

3

) x=¡ 2 §

p

3

Example 6 Self Tutor

Solve exactly forx: ¡ 3 x^2 +12x+5=0

¡ 3 x^2 +12x+5=0

) x^2 ¡ 4 x¡^53 =0 fdividing both sides by¡ 3 g

) x^2 ¡ 4 x=^53 fputting the constant on the RHSg

) x^2 ¡ 4 x+2^2 =^53 +2^2 fcompleting the squareg

) (x¡2)^2 =^173 ffactorising LHSg

) x¡2=§

q

17

3

) x=2§

q

17

3

If the coefficient of

is not , we first divide

throughout to make it.

x 2

1

1

The squared number we

add to both sides is

³

coefficient ofx

2

́ 2

If X^2 =a,

then

X=§

p

a.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_03\068CamAdd_03.cdr Thursday, 19 December 2013 3:39:12 PM GR8GREG