Solving Quadratic

Inequalities

8

9.1 Introduction

EMBAM

Now that you know howto solve quadratic equations, you are ready to move on to solving quadratic

inequalities. As with linear inequalities (whichwere covered in Grade10) your solutions willbe

intervals on the numberline, rather than single numbers.

See introductory video:VMfdy at http://www.everythingmaths.co.za

8.2 Quadratic Inequalities

A quadratic inequality is an inequality in oneof the following forms:

ax^2 + bx + c > 0

ax^2 + bx + c≥ 0

ax^2 + bx + c < 0

ax^2 + bx + c≤ 0

Solving a quadratic inequality corresponds toworking out in what region the graph of a quadratic

function lies above or below the x-axis.

Example 1: Quadratic Inequality

QUESTION

Solve the inequality 4 x^2 − 4 x + 1≤ 0 and interpret the solution graphically.

SOLUTION

Step 1 : Factorise the quadratic

Let f(x) = 4x^2 − 4 x + 1. Factorising this quadratic function gives f(x) =

(2x− 1)^2.

Step 2 : Re-write the original equation with factors