9.6 Quadratic inequalities — Learn

Prerequisites: factorising quadratics, solving quadratic equations, and testing intervals.

Objectives: solve quadratic inequalities by finding the roots, splitting the number line into intervals, and checking the sign in each interval.

This Higher section covers quadratic inequalities in one variable, within AQA reference A22.

A quadratic inequality asks where a quadratic expression is positive or negative. First find where it equals zero. These roots split the number line into intervals. The expression has a constant sign within each interval, so test one value from each.

Example 1. Solve $(x-2)(x+3)>0$.

The roots are $x=2$ and $x=-3$.

The intervals are $x<-3$, $-3<x<2$, and $x>2$.

Test $x=-4$: $(-6)(-1)>0$, so the left interval works.

Test $x=0$: $(-2)(3)<0$, so the middle interval does not work.

Test $x=3$: $(1)(6)>0$, so the right interval works.

The solution is $x<-3$ or $x>2$. The endpoints are excluded because the inequality is strict.

Example 2. Solve $(x-1)(x-5)\leq0$.

The roots are $1$ and $5$.

Test $x=3$: $(2)(-2)<0$, so values between the roots work.

Include both roots because the inequality allows equality: $1\leq x\leq5$.

Watch out. The roots alone are not the answer. Test the intervals: a positive leading coefficient is positive outside the roots and negative between them, but checking confirms the required region.

You should now be able to

• find the roots of a factorised quadratic;
• test intervals on either side of the roots;
• include or exclude roots to match the inequality sign.