Introduction to Factorising Quadratics
Factorising a quadratic expression of the form x^2 + bx + c means writing it as a product of two brackets, (x + p)(x + q), the reverse process of expanding double brackets. This works because p multiplied by q must equal c, the constant term, and p added to q must equal b, the coefficient of x.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the expression is in the form x^2 + bx + c, with the x^2 term having a coefficient of 1.
- List pairs of numbers that multiply to give c, the constant term, including negative pairs if c is negative or b is negative.
- From these pairs, find the one that also adds (keeping signs) to give b, the coefficient of x.
- Write the two brackets, (x + p)(x + q), using the pair of numbers found.
- Check the factorisation by expanding the brackets back out - it should exactly match the original quadratic.
- Recognise the special case of the difference of two squares, x^2 - a^2 = (x + a)(x - a), where there is no x term.
Worked example
Factorise x^2 + 7x + 12.
- Here b = 7 and c = 12, so two numbers are needed that multiply to give 12 and add to give 7.
- List pairs multiplying to 12: 1 and 12, 2 and 6, 3 and 4.
- Check which pair adds to 7: 3 + 4 = 7, so the pair is 3 and 4.
- Write the factorised form using this pair: (x + 3)(x + 4).
- Check by expanding: (x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12, which matches the original expression.
Practice questions
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Q1Factorise x^2 + 5x + 6.Show answer
Answer: (x + 2)(x + 3)
Q2Factorise x^2 + 9x + 20.Show answer
Answer: (x + 4)(x + 5)
Q3Factorise x^2 - 5x + 6.Show answer
Answer: (x - 2)(x - 3)
Q4Factorise x^2 + 2x - 15.Show answer
Answer: (x + 5)(x - 3)
Q5Factorise x^2 - 3x - 10.Show answer
Answer: (x - 5)(x + 2)
Q6Factorise x^2 - 16.Show answer
Answer: (x + 4)(x - 4), using the difference of two squares
Q7Factorise x^2 - x - 12.Show answer
Answer: (x - 4)(x + 3)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Factorise x^2 + 8x + 15.
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Factorise x^2 - 2x - 24.
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Factorise x^2 - 49.
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Free printable worksheet
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This topic is chapter 29 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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