Quadratic inequalities
Objective. Use roots and sign diagrams to solve quadratic inequalities.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A quadratic inequality compares a quadratic expression to zero, and is solved by first finding the roots of the related equation, then using a sketch or sign diagram to identify which regions satisfy the inequality. For example, to solve x² − 5x + 6 > 0, find the roots x = 2 and x = 3, which divide the number line into three regions; testing shows the expression is positive outside the roots, giving x < 2 or x > 3. When the inequality is "less than", the solution lies between the roots instead. Always sketch the parabola's shape to decide which region is required.
Worked example
Solve the quadratic inequality x² − x − 12 < 0, giving your answer as a range of values.
- 1Solve the related equation x² − x − 12 = 0 by factorising: (x − 4)(x + 3) = 0.
- 2The roots are x = 4 and x = −3.
- 3Since the inequality is less than zero and the parabola opens upwards, the solution lies between the roots.
- 4Solution: −3 < x < 4.
Answer: −3 < x < 4
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Practice
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Start practisingCommon mistake. Pupils write the solution to x² − 5x + 6 < 0 as x < 2 or x > 3, copying the pattern for "greater than" inequalities. Always check with the sketch: "less than zero" means between the roots.
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