Quadratic inequalities GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about quadratic inequalities

A square with a positive side length smaller than 2 has area smaller than 4. There is not just one possible side: 1, 1.5 and many others work. An inequality describes allowed values using ‘less than’ or ‘greater than’, rather than asking for one exact equality.

See the idea first

Read the height of a curve

In the algebraic question x² < 4, x may also be negative unless a context restricts it. Subtracting 4 from both sides gives x² − 4 < 0. If y = x² − 4, we want the curve's height y to be negative: below the x-axis. The symbol ≤ also allows equality; < does not.

Below the x-axis means a negative output. For x² − 4 < 0, read all x-values between −2 and 2; the zero endpoints themselves are not included.

Check one value in each region
Test x	x² − 4	Sign
-3	5	positive
0	-4	negative
3	5	positive

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a between-roots interval

What the problem asks: Solve x² − 4 < 0.

How to solve it: The roots are −2 and 2. Between them the curve is below zero, so −2 < x < 2. Exclude the endpoints because their output is zero.

Find two outside intervals

What the problem asks: Solve x² − 4 ≥ 0.

How to solve it: The curve is on or above the axis when x ≤ −2 or x ≥ 2. Use ‘or’: either separated region works.

Solve a rearranged inequality

What the problem asks: Solve x² + 2x ≤ 3.

How to solve it: Move everything to one side: x² + 2x − 3 ≤ 0. Its factors (x + 3)(x − 1) give boundaries −3 and 1. Testing x = 0 gives −3, so the solution is −3 ≤ x ≤ 1.

A reliable routine

Solve a quadratic inequality in one variable

A quadratic's sign is constant between consecutive real roots because the curve cannot move from positive to negative without passing through zero. Locate roots, then use a sketch or a test value in each region.

Rearrange so one side is zero, preserving the inequality direction.
Find every real root of the corresponding equation.
Test each region, including regions outside the extreme roots, or use the graph's shape.
Choose the required signs; include roots only for ≤ or ≥. Apply contextual restrictions last.

Check: A negative leading coefficient reverses the usual upward-parabola pattern. At a repeated root, a quadratic touches zero but keeps the same sign on both sides.

Fully worked

Quadratic inequalities GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Between roots

Higher only3 marks

Question

Solve $x^2-9<0$ .

x^2-9=(x-3)(x+3)

Boundaries: −3 and 3. At x = 0 the output is −9, so the inside region is negative. The upward curve is positive outside.

-3<x<3

Tip: −3 and 3 are excluded.

Example 2

Outside roots

Higher only3 marks

Question

Solve $x^2-x-12\ge0$ .

(x-4)(x+3)\ge0

Roots are −3 and 4. At x = 0 the product is −12, so the inside is negative. The required non-negative regions are outside, with roots included:

x\le-3\quad\text{or}\quad x\ge4

Example 3

Move the constant

Higher only4 marks

Question

Solve $x^2+3x<10$ .

Subtract 10 from both sides.

x^2+3x-10<0

(x+5)(x-2)<0

Roots: −5 and 2. At x = 0 the product is −10. The negative region is

-5<x<2

Example 4

Negative leading coefficient

Higher only4 marks

Question

Solve $6+x-x^2>0$ .

Multiply by −1 and reverse the inequality.

x^2-x-6<0

(x-3)(x+2)<0

This upward quadratic is negative between its roots.

-2<x<3

Check x = 0 in the original: 6 > 0.

Example 5

A repeated root

Higher only3 marks

Question

Solve $(x-2)^2\le0$ .

A real square is never negative. To be at most zero it must be exactly zero.

x-2=0

x=2

Here the solution is one value, not an interval.

Example 6

Context restricts the solution

Higher only4 marks

Question

A rectangle has positive width x cm and length (x + 2) cm. Its area is less than 15 cm². Find the possible widths.

x(x+2)<15

x^2+2x-15<0

(x+5)(x-3)<0

Algebraically, −5 < x < 3. But a width must be positive, so

0<x<3

The width is greater than 0 cm and less than 3 cm.

10 original questions · total 29 marks

Quadratic inequalities GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 34 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

A square limit

Higher only3 marks

Solve $x^2<16$ .

Show worked answer

Roots of x² − 16 are −4 and 4. The expression is negative between them:

-4<x<4

2

Include equality

Higher only3 marks

Solve $x^2\ge25$ .

Show worked answer

x² − 25 has roots −5 and 5 and opens upward. Choose on or above zero:

x\le-5\text{ or }x\ge5

3

Factorised inequality

Higher only3 marks

Solve $(x+1)(x-6)<0$ .

Show worked answer

Roots are −1 and 6. At x = 0 the product is −6; the upward quadratic is negative between roots.

-1<x<6

4

Positive product

Higher only3 marks

Solve $(x-2)(x-7)>0$ .

Show worked answer

The product is positive outside roots 2 and 7:

x<2\text{ or }x>7

At x = 3 the factors have opposite signs, so the middle region is not allowed.

5

Rearrange

Higher only4 marks

Solve $x^2-2x\le8$ .

Show worked answer

x^2-2x-8\le0

(x-4)(x+2)\le0

Choose the between-roots region and include equality:

-2\le x\le4

6

Downward parabola

Higher only3 marks

Solve $9-x^2\ge0$ .

Show worked answer

x^2\le9

Squared values at most 9 occur for

-3\le x\le3

This includes negative inputs.

7

No solutions

Higher only2 marks

Solve $x^2+2<0$ over the real numbers.

Show worked answer

Since $x^2\ge0$ , $x^2+2\ge2$ . It cannot be negative, so there are no real solutions.

8

All except one

Higher only3 marks

Solve $(x+3)^2>0$ .

Show worked answer

The square is positive for every real x except where it equals zero.

x+3=0\Rightarrow x=-3

Answer: all real x except −3, or x < −3 or x > −3.

9

Integer solutions

Higher only3 marks

List the integer solutions of $x^2+x-6<0$ .

Show worked answer

(x+3)(x-2)<0

Hence −3 < x < 2. The integers strictly inside are −2, −1, 0, 1.

10

Do not stop at the roots

Higher only2 marks

A student solves $x^2-1<0$ and answers x = −1 or 1. Explain the error and correct it.

Show worked answer

Those values make the output zero, so they fail the strict inequality. Between the roots the output is negative:

-1<x<1

Examiner-style feedback

Common quadratic inequalities mistakes

Giving just the roots

Use roots as boundaries, then choose the regions with the required sign.

Using ‘and’ for separate regions

x < −2 or x > 3 means either region works. No number can satisfy both conditions simultaneously.

Forgetting context

A negative algebraic solution may be invalid for a length, time or count.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Make one side zero.
Find boundaries.
Check signs in every region.
Include endpoints only when allowed.

Quick answers

Quadratic inequalities FAQ

Can I always shade between the roots?

No. The requested sign and whether the parabola opens up or down determine the regions.

What if there are no real roots?

The sign does not change anywhere. The inequality may hold for every real value or for none; inspect a test value or the square form.

Build connected skills

What to revise next

Read a curve

Quadratic graphs

Connect signs to above and below the axis.

Revise quadratic graphs

Find boundaries

Factorising

Find roots using brackets.

Revise factorising

Content standards

Curriculum and rights review

A22 quadratic inequalities in one real variable: Higher tier. Includes repeated roots, no real roots and contextual restrictions. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content