Quadratic Inequalities

Key idea

Find where a parabola is above or below the axis.

In depth

Find the roots, which split the number line into regions, then test a point in each region to see where the inequality holds. The parabola is above the axis outside the roots (if it opens up) and below it between them.

Worked example

x² − 4 < 0: roots ±2; the parabola is below the axis between them, so −2 < x < 2.

Real-world uses

• Finding the range of prices that keep profit positive
• Safe operating ranges in engineering

Common mistake

Dividing by an expression with an unknown sign, which can flip the inequality.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related advanced topics in Algebra

• Absolute Value Inequalities – Rewrite as a compound inequality.
• Systems with Three Variables – Reduce to two variables then solve in sequence.
• Systems of Inequalities – Solution region is the overlap of half-planes.
• Factoring by Grouping – Group terms in pairs to find a common binomial.
• Sum and Difference of Cubes – Special factoring patterns for cubes.
• Completing the Square – Rewrite a quadratic as a perfect square plus constant.