To solve a quadratic inequality, find the roots of the quadratic, then sketch its U-shaped graph. The curve is below the x-axis between the roots and above it outside them. So an inequality with < 0 or ≤ 0 has a solution between the roots, and one with > 0 or ≥ 0 has a solution outside the roots, written as two separate inequalities.
Quadratic inequalities
- Rearrange so one side is 0.
- Factorise to find the roots.
- Sketch the U-shaped curve through the roots.
- < 0: between the roots. > 0: outside the roots.
With ≤ or ≥, include the roots themselves.
Why a sketch helps
A quadratic inequality asks where the graph of the quadratic is above or below the x-axis. Because a quadratic with a positive x² term is U-shaped, it dips below the axis between its two roots and rises above it outside them. A quick sketch, showing just the roots and the U shape, makes the correct region obvious.
Finding the roots
Rearrange so that one side is zero, then factorise. x² − 3x − 10 = (x − 5)(x + 2), so the roots are x = 5 and x = −2. If the quadratic does not factorise, use the formula to find the roots.
Choosing the region
If the inequality is < 0 or ≤ 0, you want the part of the curve below the axis, which is between the roots: −2 < x < 5. If it is > 0 or ≥ 0, you want the parts above the axis, outside the roots: x < −2 or x > 5. The second answer must be written as two inequalities, never as a single combined one.
Rearranging first
If both sides contain terms, bring everything to one side before factorising. x² > 3x + 4 becomes x² − 3x − 4 > 0, which factorises to (x − 4)(x + 1) > 0, giving x < −1 or x > 4.
Listing integer solutions
To list the integers that satisfy an inequality, solve it first and then pick out the whole numbers in the solution. Pay attention to whether the end values are included: for −1 < x < 5, the integers are 0, 1, 2, 3 and 4.
Worked examples
Example 1. Solve x² − 3x − 10 > 0.
- Factorise: (x − 5)(x + 2), so the roots are 5 and −2.
- > 0 means above the axis: outside the roots.
Answer: x < −2 or x > 5.
Example 2. Solve x² + x − 12 ≤ 0.
- Factorise: (x + 4)(x − 3), so the roots are −4 and 3.
- ≤ 0 means below or on the axis: between the roots, including them.
Answer: −4 ≤ x ≤ 3.
Example 3. Solve x² − 49 < 0.
- The roots are 7 and −7.
- < 0: between the roots.
Answer: −7 < x < 7.
Example 4. List the integers that satisfy x² − 4x − 5 < 0.
- Factorise: (x − 5)(x + 1), so −1 < x < 5.
- Integers strictly between −1 and 5.
Answer: 0, 1, 2, 3, 4.
Example 5. Solve x² > 3x + 4.
- Rearrange: x² − 3x − 4 > 0, which is (x − 4)(x + 1) > 0.
- > 0: outside the roots −1 and 4.
Answer: x < −1 or x > 4.
Common mistakes
- Choosing the wrong region. Sketch the curve: < 0 is between the roots for a U shape.
- Writing x < −2 and x > 5 as one inequality. Write them separately, joined by “or”.
- Not rearranging to zero. Bring all terms to one side before factorising.
- Including or excluding the roots wrongly. ≤ and ≥ include the roots; < and > do not.
- Square-rooting both sides of x² < 49. This loses the negative values; the answer is −7 < x < 7.
How quadratic inequalities are examined
Quadratic inequalities are a Higher tier topic in Edexcel GCSE (1MA1) and the International GCSE (4MA1). Questions usually ask you to:
- factorise a quadratic, then solve the inequality
- solve an inequality of the form x² < k or x² > k
- list the integers that satisfy a quadratic inequality
- rearrange an inequality before solving
The inequalities worksheet includes Higher questions with answers, and the GCSE algebra revision engine gives more practice.
This combines solving quadratics with linear inequalities. At A-Level, see inequalities.
Frequently asked questions
What is the quickest way to solve a quadratic inequality?
Find the roots, sketch the U-shaped graph, then choose the region between or outside the roots.
When is the answer between the roots?
When the inequality is < 0 or ≤ 0, for a quadratic with a positive x² term.
Why do some answers have two parts?
Because the curve is above the axis on both sides of the roots, giving two separate regions.