Cambridge International AS and A Level Mathematics Pure Mathematics 1

Equations that cannot be factorised

P1^

1

x 0 1 2 3 4 5 6

x^20149162536

−6x 0 −6 −12 −18 −24 −30 −36

+2 +2 +2 +2 +2 +2 +2 +2

y +2 −3 −6 −7 −6 −3 +2

From figure 1.6, x is between 0.3 and 0.4 so approximately 0.35, or between 5.6
and 5.7 so approximately 5.65.

Clearly the accuracy of the answer is dependent on the scale of the graph but,
however large a scale you use, your answer will never be completely accurate.

Completing the square

If a quadratic equation has a solution, this method will give it accurately. It
involves adjusting the left-hand side of the equation to make it a perfect square.
The steps involved are shown in the following example.

0

1 2 3 4 5 6

1

–1

–2

–3

–4

–5

–6

–7

2

y

x

Figure 1.6

Between

0.3 and 0.4

Between

5.6 and 5.7