All subjects · Mathematics · WMA11 (P1)

Quadratic inequalities

To solve a quadratic inequality, find where the quadratic equals zero, then use a sketch to see where it is above or below the x-axis.

Method

1. Rearrange so one side is 0.

2. Solve the equation to find the critical values.

3. Sketch the graph.

4. Read off the x-values where the graph is above (> 0) or below (< 0) the x-axis.

Reading the answer

For a u-shaped graph with roots p < q:

Write two intervals with 'or' or with ∪, never as one double inequality.

Traps

If x2 has a negative coefficient, multiply through by -1 and reverse the sign first. Do not divide by an expression containing x. Questions on the discriminant often lead to a quadratic inequality in k.

Worked example

Solve x2 - x - 6 > 0.

  1. (x - 3)(x + 2) = 0 gives critical values -2 and 3.
  2. The graph is u-shaped, so it is above the x-axis outside the roots.

Answer: x < -2 or x > 3

Key idea: Find critical values, sketch, then read off: between the roots for < 0, outside the roots for > 0 (when x2 is positive).

Check yourself

  1. Solve x2 - 5x + 4 < 0.
    • x < 1 or x > 4
    • -4 < x < -1
    • 1 < x < 4
    • x < 4
    Show answer

    1 < x < 4. (x - 1)(x - 4) < 0. The u-shaped graph is below the axis between the roots.

  2. Solve x2 ≥ 9.
    • x ≤ -3 or x ≥ 3
    • x ≥ 3
    • -3 ≤ x ≤ 3
    • x ≤ -3 and x ≥ 3
    Show answer

    x ≤ -3 or x ≥ 3. x2 - 9 ≥ 0, so (x - 3)(x + 3) ≥ 0: outside the roots.

  3. Solve 6 + x - x2 > 0.
    • x < -2 or x > 3
    • -3 < x < 2
    • -2 < x < 3
    • x > 3
    Show answer

    -2 < x < 3. Multiply by -1 and reverse: x2 - x - 6 < 0, so (x - 3)(x + 2) < 0, giving -2 < x < 3.

  4. x2 + kx + 4 = 0 has no real roots. Find the set of possible values of k.
    • k < 4
    • -4 < k < 4
    • k < -4 or k > 4
    • k > 4
    Show answer

    -4 < k < 4. b2 - 4ac < 0 gives k2 - 16 < 0, so -4 < k < 4.

  5. How many integers satisfy x2 - 2x - 15 ≤ 0?
    Show answer

    9. (x - 5)(x + 3) ≤ 0 gives -3 ≤ x ≤ 5. The integers -3 to 5 inclusive are 9 numbers.

  6. Find the values of x that satisfy both x2 - 4x - 12 < 0 and 2x - 3 > 5.
    • -2 < x < 6
    • 4 < x < 6
    • x > 4
    • -2 < x < 4
    Show answer

    4 < x < 6. (x - 6)(x + 2) < 0 gives -2 < x < 6. The linear one gives x > 4. The overlap is 4 < x < 6.

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Last updated 7 November 2026. Spotted a mistake? Tell us.