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Quadratic Inequalities

HigherHigher tier onlyAQAEdexcelOCR

Practise quadratic inequalities with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through solving quadratic inequalities, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Sketch the parabola and read where it's above or below the x-axis.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A quadratic inequality asks for the range of values satisfying an expression that is greater or less than zero, rather than equal to it.

The method starts the same way: factorise and find the critical values where the expression equals zero. Those values divide the number line into regions.

The decisive step is deciding which regions satisfy the inequality, and a sketch makes this easy. For a positive \(x^2\) coefficient, the curve is below the axis between the roots and above it outside them. So less than zero gives a single range between the roots, and greater than zero gives two separate ranges.

Revision notes

Finding the critical values

Set the expression equal to zero and solve, usually by factorising.

For \(x^2 - x - 6 < 0\), factorising gives \((x - 3)(x + 2) = 0\), so the critical values are \(3\) and \(-2\).

Using a sketch to decide

Sketch the parabola through the critical values. With a positive \(x^2\) term, the curve dips below the axis between the roots.

Since the inequality asks for less than zero, the answer is the region between them: \(-2 < x < 3\).

Greater than zero

Above the axis means outside the roots, which needs two separate statements.

For \(x^2 - x - 6 > 0\), the answer is \(x < -2\) or \(x > 3\). Writing this as a single double inequality would be wrong, since the two regions are not joined.

Key points

  • Factorise to find the critical values.
  • The critical values divide the number line into regions.
  • Sketch the parabola to decide which regions apply.
  • Less than zero gives the region between the roots.
  • Greater than zero gives two separate regions.
  • Use \(\le\) or \(\ge\) if the inequality includes equality.

Worked examples

Example 1

Solve \(x^2 - 4 < 0\).

Working

\[(x - 2)(x + 2) = 0 \Rightarrow x = \pm 2\]find the critical values
\[\text{Curve is below the axis between the roots}\]sketch to decide the region
\[-2 < x < 2\]state the solution

Example 2

Solve \(x^2 - 5x + 6 > 0\).

Working

\[(x - 2)(x - 3) = 0 \Rightarrow x = 2, 3\]find the critical values
\[\text{Above the axis outside the roots}\]sketch to decide the regions
\[x < 2 \text{ or } x > 3\]state both ranges

Example 3

Solve \(x^2 + 2x - 8 \le 0\).

Working

\[(x + 4)(x - 2) = 0 \Rightarrow x = -4, 2\]find the critical values
\[\text{Between the roots, including them}\]the inequality includes equality
\[-4 \le x \le 2\]state the solution

Common mistakes

  • Giving the critical values as the answer.

    The solution is a range of values, not the two roots themselves.

  • Writing two separate regions as one double inequality.

    x < 2 or x > 3 cannot be written as 3 < x < 2, which means nothing.

  • Getting the region the wrong way round.

    A quick sketch settles it: below the axis is between the roots for a positive x² term.

  • Using strict inequalities when equality is included.

    ≤ requires the endpoints to be included in the answer.

Exam tips

  • Always sketch the parabola — it removes the guesswork.
  • Mark the critical values on the sketch before deciding.
  • Use or between two separate regions.
  • Check a test value from your range in the original inequality.

Key terms

Critical value
A value where the expression equals zero.
Quadratic inequality
An inequality involving an \(x^2\) term.
Region
A section of the number line satisfying the inequality.
Parabola
The curve of a quadratic function.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.