Cambridge International AS and A Level Mathematics Pure Mathematics 1

The quadratic formula

P1^

1

4 The curves below all have equations of the form y = x^2 + bx + c.

In each case find the values of b and c.

5 Solve the following equations by completing the square.

(i) x^2 − 6 x + 3 = 0 (ii) x^2 − 8 x – 1 = 0

(iii) x^2 − 3 x + 1 = 0 (iv) 2 x^2 − 6 x + 1 = 0

(v) 5 x^2 + 4 x − 2 = 0

The quadratic formula

Completing the square is a powerful method because it can be used on any

quadratic equation. However it is seldom used to solve an equation in practice

because it can be generalised to give a formula which is used instead. The

derivation of this follows exactly the same steps.

To solve a general quadratic equation ax^2 + bx + c = 0 by completing the square:

First divide both sides by a: ⇒ x^2 ++bxa ca= 0.

Subtract the constant term from both sides of the equation:

⇒ x^2 +=bxa −ac

y

x

(3, 1)

(i) y

x

(–1, –1)

(ii)

y

4  x

(iii) y

x

(–3, 2)

(iv)