Sheet № 141 · Higher only · AQA · Edexcel · OCR
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 AQ
§Key definitions
Question:
Solve x² − 9 < 0.
Answer:
−3 < x < 3
Q1 (Foundation):
Solve x² − 16 > 0.
Q2 (Higher):
Solve x² − 7x + 10 ≤ 0.
Q3 (Higher):
Solve 3x² − x − 2 > 0.
§Formulas to memorise
Factorise the quadratic, find the roots, then sketch the parabola to identify solution intervals
For x² > 0 shaped parabola: below the x-axis between roots, above outside the roots
Rearrange the inequality — so that one side is zero (e.g., x² − 5x + 6 < 0).
Factorise the quadratic — (or use the quadratic formula to find roots).
Find the roots — these are the x-values where the quadratic equals zero.
Sketch the parabola. — If the coefficient of x² is positive, the parabola is U-shaped. If negative, it is ∩-shaped.
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.
Write the solution — using inequality notation.
Worked example
Solve x² − 9 < 0.
Working:
⚠ 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 >).