Cambridge International AS and A Level Mathematics Pure Mathematics 1

Algebra

22

P1^

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EXAMPLE 1.30 Solve the equation x^2 − 6 x + 2 = 0 by completing the square.

SOLUTION

Subtract the constant term from both sides of the equation:

⇒ x^2 − 6 x = − 2

Take the coefficient of x : − 6

Halve it: − 3

Square the answer: + 9

Add it to both sides of the equation:

⇒ x^2 − 6 x + 9 = − 2 + 9

Factorise the left-hand side. It will be found to be a perfect square:

⇒ (x − 3)^2 = 7

Take the square root of both sides:

⇒ x − 3 = ± 7

⇒ x = 3 ± 7

Using your calculator to find the value of 7

⇒ x = 5.646 or 0.354, to 3 decimal places.

The graphs of quadratic functions

Look at the curve in figure 1.7. It is the graph of y = x^2 − 4 x + 5 and it has the

characteristic shape of a quadratic; it is a parabola.

Notice that:

●●it has a minimum point

(or vertex) at (2, 1)

●●●it has a line of symmetry, x = 2.

It is possible to find the vertex

and the line of symmetry without

plotting the points by using the

technique of completing the

square.

} left-hand side a perfect square.●? Explain why this makes the

This is an exact answer.

This is an approximate answer.

–   2 3 4



2

3

4



y

x

x = 2

(2)

Figure 1.7