Quadratic inequalities

The method

1. Rearrange so the right-hand side is zero. Move every term to the left so you have ax² + bx + c compared with 0.
2. Find the critical values. Solve ax² + bx + c = 0 by factorising (or the formula).
3. Sketch the graph. When the coefficient of x² is positive, the parabola opens upwards and crosses the x-axis at the critical values.
4. Read off the region. “< 0” is the part below the x-axis (between the roots). “> 0” is above (outside the roots, written as two inequalities).

Mistakes that lose marks

• Writing x < 5 and x < −1.5 after finding the roots, instead of thinking about the graph.
• Writing “−1.5 > x > 5”, which no number satisfies. For two regions write “x < −1.5 or x > 5”.
• Using ≤ when the question has < (or the other way round). The roots are included only for ≤ and ≥.
• Dividing both sides by x, which can lose a solution and flips the sign when x is negative.

Six exam-style questions

These are our own questions in the style of the exam, not past-paper questions. In every question x, n and k are real numbers (n an integer in question 4). Try each one, then open the solution.

1. EXAM-STYLE Solve x² − 5x + 6 > 0 [3 marks]

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   1. Factorise: (x − 2)(x − 3) > 0
   2. Critical values: x = 2 and x = 3
   3. The coefficient of x² is positive, so y = x² − 5x + 6 opens upwards and is above the x-axis outside the roots
   4. x < 2 or x > 3

2. EXAM-STYLE Solve 3x² + 4x − 4 ≤ 0 [3 marks]

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   1. Factorise: (3x − 2)(x + 2) ≤ 0
   2. Critical values: x = ⅔ and x = −2
   3. The parabola opens upwards, so it is on or below the x-axis between the roots, roots included
   4. −2 ≤ x ≤ ⅔

3. EXAM-STYLE Solve x² < 4x + 12 [3 marks]

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   1. Rearrange so the right-hand side is zero: x² − 4x − 12 < 0
   2. Factorise: (x − 6)(x + 2) < 0
   3. Critical values −2 and 6; the parabola is below the x-axis between them
   4. −2 < x < 6

4. EXAM-STYLE Find all the integers n that satisfy n² − n − 20 < 0 [3 marks]

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   1. Factorise: (n − 5)(n + 4) < 0, so −4 < n < 5
   2. The inequality is strict, so −4 and 5 are not included
   3. n = −3, −2, −1, 0, 1, 2, 3, 4 (8 integers)

5. EXAM-STYLE Solve 2x² + 3x > 5 [3 marks]

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   1. Rearrange so the right-hand side is zero: 2x² + 3x − 5 > 0
   2. Factorise: (2x + 5)(x − 1) > 0, critical values −5/2 and 1
   3. The parabola opens upwards, so it is above the x-axis outside the roots
   4. x < −5/2 or x > 1

6. EXAM-STYLE The equation x² + kx + 9 = 0 has two distinct real roots. Find the set of possible values of k. (Additional / Further Maths) [4 marks]

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   1. Two distinct real roots means the discriminant is positive: b² − 4ac > 0
   2. k² − 4(1)(9) > 0, so k² − 36 > 0
   3. (k − 6)(k + 6) > 0, critical values ±6
   4. k < −6 or k > 6

Where this is in the syllabus

Quadratic inequalities appear in Cambridge IGCSE Mathematics 0580 (Extended), Pearson Edexcel International GCSE Mathematics A 4MA1 (Higher) and GCSE Higher tier. For the official specification and past papers, use the board’s own website: Cambridge 0580 and Edexcel 4MA1.