Quadratic Inequalities –

Quadratic inequalities take the skills you have learned from solving quadratic equations and combine them with your understanding of inequalities. Instead of finding exact values where a quadratic equals zero, you find the range of values where the quadratic is greater than or less than zero. This is a Higher-only topic that appears on AQA, Edexcel, and OCR papers and often carries 3–4 marks.

What Are Quadratic Inequalities?

A quadratic inequality looks like x² − 5x + 6 < 0 or 2x² + 3x − 5 ≥ 0. You need to find the set of x-values that make the inequality true. The solution is typically an interval (or a pair of intervals) rather than a single value.

Key Formulas

Step-by-Step Method

1. Rearrange the inequality so that one side is zero (e.g., x² − 5x + 6 < 0).
2. Factorise the quadratic (or use the quadratic formula to find roots).
3. Find the roots — these are the x-values where the quadratic equals zero.
4. Sketch the parabola. If the coefficient of x² is positive, the parabola is U-shaped. If negative, it is ∩-shaped.
5. Read the solution from the sketch. For < 0, you want where the curve is below the x-axis. For > 0, you want where it is above the x-axis.
6. Write the solution using inequality notation.

Worked Example 1 — Foundation Level

Question: Solve x² − 9 < 0.

Working:

Factorise: x² − 9 = (x + 3)(x − 3) = 0 gives roots x = −3 and x = 3.

The coefficient of x² is positive, so the parabola is U-shaped. The curve is below the x-axis between the roots.

So x² − 9 < 0 when −3 < x < 3.

Answer: −3 < x < 3

Worked Example 2 — Higher Level

Question: Solve x² − 5x + 6 < 0.

Working:

Factorise: x² − 5x + 6 = (x − 2)(x − 3) = 0 gives roots x = 2 and x = 3.

Positive x² means U-shaped parabola. The curve is below the x-axis between the two roots.

So x² − 5x + 6 < 0 when 2 < x < 3.

Check: try x = 2.5: (2.5)² − 5(2.5) + 6 = 6.25 − 12.5 + 6 = −0.25 < 0 ✓

Answer: 2 < x < 3

Worked Example 3 — Exam Style

Question: Solve 2x² + x − 6 ≥ 0.

Working:

Factorise using the ac method. a = 2, b = 1, c = −6. ac = −12. Two numbers that multiply to −12 and add to 1: 4 and −3.

2x² + 4x − 3x − 6 = 2x(x + 2) − 3(x + 2) = (x + 2)(2x − 3) = 0.

Roots: x = −2 and x = 3/2.

Positive x² means U-shaped. The curve is above or on the x-axis outside and at the roots.

So 2x² + x − 6 ≥ 0 when x ≤ −2 or x ≥ 3/2.

Check: try x = 0: 0 + 0 − 6 = −6 < 0 (between roots, not included) ✓ Try x = 2: 8 + 2 − 6 = 4 ≥ 0 ✓

Answer: x ≤ −2 or x ≥ 3/2

Common Mistakes

• Writing the answer as a single inequality. For "greater than" inequalities, the solution is two separate regions (x ≤ a or x ≥ b), not a ≤ x ≤ b. The parabola is above the x-axis on both sides of the roots.
• Forgetting to sketch the parabola. Without a sketch, students often guess the wrong region. A quick sketch takes seconds and prevents errors.
• Mixing up < and >. If you want where the quadratic is negative, look below the x-axis. If positive, look above. The sketch makes this clear.

Exam Tips

• Always draw a quick sketch of the parabola — it does not need to be accurate, just the right shape with roots marked.
• Use a test value between the roots and outside the roots to verify your answer.
• If the coefficient of x² is negative, the parabola is ∩-shaped, which reverses the regions.
• For ≤ and ≥, include the roots in your answer with ≤ or ≥ (not strict < or >).

Practice Questions

Q1 (Foundation): Solve x² − 16 > 0.

Q2 (Higher): Solve x² − 7x + 10 ≤ 0.

Q3 (Higher): Solve 3x² − x − 2 > 0.

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Related Topics

• Solving Quadratic Equations by Factorising
• Solving Inequalities
• Quadratic Graphs

Summary

• Quadratic inequalities ask for the range of x-values where a quadratic is positive or negative.
• Factorise to find the roots, then sketch the parabola to determine the correct region.
• For a positive x² coefficient: the quadratic is negative between the roots and positive outside them.
• Write your answer as an inequality — use "and" (between roots) or "or" (outside roots).
• A quick sketch and a test value are the best ways to avoid mistakes.