Quadratic Inequalities
A quadratic inequality compares a quadratic expression to zero, such as x^2 - 5x + 6 > 0, and its solution is a range (or two ranges) of x-values rather than a single number. It is solved by factorising to find the critical values where the expression equals zero, then sketching the parabola to decide which region(s) satisfy the inequality. This extends GCSE quadratic skills into AQA Further Maths.
Before you start
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Method
- Rearrange the inequality so that zero is on one side, leaving a quadratic expression compared to zero.
- Factorise the quadratic (or use the quadratic formula) to find its critical values, the x-values where the expression equals zero.
- Sketch the parabola using the critical values as its x-intercepts, noting whether it opens upwards or downwards from the sign of the x^2 coefficient.
- Use the sketch to identify which x-values make the expression positive (above the x-axis) or negative (below the x-axis).
- Write the solution as an inequality (or two separate inequalities) using the critical values, taking care with strict and non-strict inequality signs.
- If you multiply or divide the inequality by a negative number at any point, remember to reverse the inequality sign.
Worked example
Solve the inequality x^2 - 3x - 10 < 0.
- Rearrange so zero is on one side (it already is): x^2 - 3x - 10 < 0.
- Factorise the quadratic: (x - 5)(x + 2) < 0.
- Find the critical values by setting each factor to zero: x = 5 and x = -2.
- Sketch the parabola: it opens upwards and crosses the x-axis at x = -2 and x = 5, so it is below the x-axis between these values.
- Final answer: -2 < x < 5.
Practice questions
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Q1Solve the inequality x^2 - 25 > 0.Show answer
Answer: x < -5 or x > 5
Q2Solve the inequality x^2 - 8x + 15 <= 0.Show answer
Answer: 3 <= x <= 5
Q3Solve the inequality 2x^2 - 5x - 3 >= 0.Show answer
Answer: x <= -1/2 or x >= 3
Q4Solve the inequality x^2 + 3 < 4x.Show answer
Answer: 1 < x < 3 (rearrange first to x^2 - 4x + 3 < 0)
Q5Solve the inequality 36 - x^2 >= 0.Show answer
Answer: -6 <= x <= 6
Q6Solve the inequality x^2 + x - 12 <= 0. Hence write down all the integer values of x that satisfy it.Show answer
Answer: -4 <= x <= 3; integers -4, -3, -2, -1, 0, 1, 2, 3
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Solve the inequality x^2 - 81 > 0.
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A rectangular garden has length (x + 5) metres and width (x - 3) metres, where x is greater than 3. The area of the garden is greater than 48 m^2. Form and solve an inequality to find the range of possible values of x.
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Find the set of values of k for which the equation x^2 + kx + (k + 8) = 0 has no real roots.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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