Cambridge International AS and A Level Mathematics Pure Mathematics 1

The quadratic formula

P1^

1

Substituting these values in x bb ac
a

=––±

(^24)
2
gives^2428
24
2

±

±

–

= –

24 8

2

24

2

±

±

–

–

Trying to find the square root of a negative number creates problems.
A positive number multiplied by itself is positive: +2 × + 2 = +4.
A negative number multiplied by itself is also positive: − 2 × − 2 = +4.
Since − 4 can be neither positive nor negative, no such number exists, and so
you can find no real solution.

Note

It is not quite true to say that a negative number has no square root. Certainly it
has none among the real numbers but mathematicians have invented an imaginary
number, denoted by i, with the property that i^2 = −1. Numbers like 1 + i and −1 − i
(which are in fact the solutions of the equation above) are called complex numbers.
Complex numbers are extremely useful in both pure and applied mathematics; they
are covered in P3.

To return to the problem of solving the equation x^2 − 2 x + 2 = 0, look what
happens if you draw the graph of y = x^2 − 2 x + 2. The table of values is given
below and the graph is shown in figure 1.9. As you can see, the graph does not
cut the x axis and so there is indeed no real solution to this equation.

x − 1 0 1 2 3

x^2 + 1 0 + 1 + 4 + 9

− 2 x + 2 0 –2 − 4 − 6

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

y + 5 + 2 + 1 + 2 + 5

–1 0 1 2 3

–1

1

2

3

4

5

y

x

Figure 1.9