Cambridge International AS and A Level Mathematics Pure Mathematics 1

Algebra

P1^

1

Quadratic inequalities

EXAMPLE 1.39 Solve (i) x^2 − 4 x + 3  0 (ii) x^2 − 4 x + 3  0.

SOLUTION

The graph of y = x^2 − 4 x + 3 is shown in figure 1.14 with the green parts of the

x axis corresponding to the solutions to the two parts of the question.

(i) You want the values of x for which (ii) You want the values of x for

y  0, which that is where the curve y  0, that is where the curve

is above the x axis. crosses or is below the x axis.

The solution is x  1 or x  3. The solution is x  1 and x  3,

usually witten 1  x  3.

EXAMPLE 1.40 Find the set of values of k for which x^2 + kx + 4 = 0 has real roots.

SOLUTION

A quadratic equation, ax^2 + bx + c = 0 , has real roots if b^2 − 4 ac  0.

So x^2 + kx + 4 = 0 has real roots if k^2 − 4 × 4 × 1  0.

⇒ k^2 − 16  0

⇒ k^2  16

So the set of values is k  4 and k  −4.

Here the end points are not included in the

inequality so you draw open circles:

Here the end points are included in the

inequality so you draw solid circles: •

0 1 2 3 4 x

–1

2

1

3

y

0 1 2 3 4 x

–1

2

1

3

y

Figure 1.14

Take the square root

of both sides.

Take care: (–5)^2 = 25 and

(–3)^2 = 9, so k must be

less than or equal to –4.