Skip to content
VirtusAcademy

Factorising Quadratics

FoundationHigherAQAEdexcelOCR

Factorising Quadratics is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on factorising quadratic expressions, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Find two numbers that multiply to the constant and add to the x-coefficient.

Free downloads

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.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Factorising a quadratic turns \(x^2 + bx + c\) into two brackets. You need two numbers that multiply to give \(c\) and add to give \(b\).

For \(x^2 + 7x + 12\), the pair is \(3\) and \(4\), since \(3 \times 4 = 12\) and \(3 + 4 = 7\). The factorised form is \((x + 3)(x + 4)\).

Signs follow a reliable pattern. If \(c\) is positive, both numbers share the sign of \(b\). If \(c\) is negative, the numbers have opposite signs, and the larger one takes the sign of \(b\). Working through the factor pairs of \(c\) systematically finds the right combination quickly.

Revision notes

Finding the pair

List the factor pairs of the constant, then check which pair adds to the coefficient of \(x\).

For \(x^2 + 9x + 20\): the pairs of \(20\) are \(1 \times 20\), \(2 \times 10\) and \(4 \times 5\). Only \(4 + 5 = 9\), so the answer is \((x + 4)(x + 5)\).

Handling negative signs

If the constant is negative, the two numbers have opposite signs and their difference gives the coefficient of \(x\).

For \(x^2 - 2x - 15\): the pair is \(-5\) and \(+3\), since \(-5 \times 3 = -15\) and \(-5 + 3 = -2\), giving \((x - 5)(x + 3)\).

The difference of two squares

An expression of the form \(x^2 - a^2\) factorises to \((x + a)(x - a)\), with no middle term.

So \(x^2 - 36 = (x + 6)(x - 6)\). Recognising this pattern saves searching for a pair that adds to zero.

Key points

  • Find two numbers multiplying to \(c\) and adding to \(b\).
  • Both numbers share the sign of \(b\) when \(c\) is positive.
  • The numbers have opposite signs when \(c\) is negative.
  • List the factor pairs of \(c\) systematically.
  • \(x^2 - a^2\) factorises to \((x+a)(x-a)\).
  • Check by expanding your brackets.

Worked examples

Example 1

Factorise \(x^2 + 8x + 15\).

Working

\[3 \times 5 = 15 \text{ and } 3 + 5 = 8\]find the pair multiplying to 15 and adding to 8
\[(x + 3)(x + 5)\]write the two brackets

Example 2

Factorise \(x^2 - 3x - 10\).

Working

\[-5 \times 2 = -10 \text{ and } -5 + 2 = -3\]the constant is negative, so the signs differ
\[(x - 5)(x + 2)\]write the two brackets

Example 3

Factorise \(x^2 - 49\).

Working

\[x^2 - 7^2\]recognise the difference of two squares
\[(x + 7)(x - 7)\]write the factorised form

Common mistakes

  • Finding a pair that adds correctly but multiplies wrongly.

    Both conditions must hold. Check the product as well as the sum.

  • Getting the signs the wrong way round.

    For x² − 2x − 15 the pair is −5 and +3, not +5 and −3, since the sum must be −2.

  • Missing a difference of two squares.

    x² − 36 has no middle term and factorises to (x + 6)(x − 6).

  • Not checking by expanding.

    Multiplying the brackets back out catches sign errors immediately.

Exam tips

  • List the factor pairs of the constant before guessing.
  • Use the sign rules to narrow down the possibilities.
  • Look out for the difference of two squares.
  • Always expand your answer to verify it.

Key terms

Quadratic
An expression whose highest power is \(x^2\).
Constant
The term with no variable, here \(c\).
Difference of two squares
An expression of the form \(x^2 - a^2\).
Factorise
To write as a product of brackets.

Related topics

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