3.5PureStretch

Quadratic Inequalities

A quadratic inequality asks for the values of $x$ where a parabola is above or below the axis. The reliable method is: find the roots, sketch the parabola, then read off the region the inequality describes.

30 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.5 Quadratic InequalitiesWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Find the critical values (roots) of a quadratic
Use a sketch to identify the correct region
Solve inequalities of the form ax² + bx + c > 0 (and < 0)
Express the solution set correctly

Step 1 — find the critical values

Treat the quadratic as an equation first and solve $a x^{2} + b x + c = 0$ . These roots are the $critical values$ where the curve crosses the $x$ -axis.

1Solve

x^{2} - x - 6 = 0

2Factorise:

(x - 3) (x + 2) = 0

, so

x = 3

x = - 2

Answercritical values

x = - 2

and

x = 3

Step 2 — sketch and read the region

Sketch the parabola through the roots. For $> 0$ you want where the curve is $above$ the axis; for $< 0$ where it is $below$ . A $\cup$ parabola is above the axis outside the roots and below between them.

(x + 2) (x - 3) > 0 \Rightarrow x < - 2 or x > 3

Above the axis means outside the roots (for a positive

x^{2}

coefficient).

1Roots at

- 2

and

3

; the

\cup

curve is below the axis between them.

\leq 0

includes the roots themselves.

Answer

- 2 \leq x \leq 3

Tip — “Less than 0” → between the roots; “greater than 0” → outside the roots (for a ∪ parabola).

Writing the answer

A “between” region is a single inequality like $- 2 \leq x \leq 3$ . An “outside” region needs two parts joined by “or”: $x < - 2$ or $x > 3$ .

Formula recap

a x^{2} + b x + c = 0 \Rightarrow critical values

Find the roots first.

> 0 : outside roots, < 0 : between roots

For a ∪ parabola.

x < - 2 or x > 3

“Outside” needs two parts.

Common mistakes to avoid

Writing the “outside” region as a single double inequality like 3 < x < −2.

Use two separate parts joined by “or”: x < −2 or x > 3.

Guessing the region without a sketch.

A quick parabola sketch makes the correct region obvious.

Key takeaways

Find the critical values by solving the quadratic equation.
Sketch the parabola to see where it is above or below the axis.
< 0 → between the roots; > 0 → outside the roots (for a ∪ curve).
Outside regions need two parts joined by “or”.

Test yourself

Ready to lock in Quadratic Inequalities? Pick a mode and earn XP & Dobloons.