16.4 Factorising Quadratic Expressions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: common factors, factorising x² + bx + c into two brackets, and the difference of two squares. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Factorise by taking out a common factor, such as x² + 7x = x(x + 7)
• Factorise quadratic expressions of the form x² + bx + c
• Factorise the difference of two squares, such as x² − 49
• Check a factorisation by expanding the brackets

Key words

• Factorise: Write an expression as a product of factors, usually with brackets. It is the reverse of expanding.
• Factor pair: Two numbers that multiply to give a number, e.g. 3 and 4 for 12.
• Coefficient: The number in front of a letter. In x² + 7x + 12 the coefficient of x is 7.
• Difference of two squares: An expression like x² − 49, one square subtracted from another.

Taking out a common factor

If every term shares a factor, take it outside a single bracket. Take out the highest common factor, including any x.

Worked example: single brackets

x² + 7x: both terms contain x, so x² + 7x = x(x + 7).
3x² − 12x: the HCF of 3x² and 12x is 3x, so 3x² − 12x = 3x(x − 4).

Answer: x(x + 7) and 3x(x − 4)

Factorising x² + bx + c

From section 16.1, (x + p)(x + q) = x² + (p + q)x + pq. So to factorise x² + bx + c, find two numbers that multiply to c and add to b.

Worked example: x² + 7x + 12

Factor pairs of 12: 1 and 12, 2 and 6, 3 and 4.
Which pair adds to 7? 3 + 4 = 7.

Answer: (x + 3)(x + 4)

Key fact: Check by expanding: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓

Factorising with negative numbers

• x² + bx + c; Signs of the two numbers both positive; Example x² + 9x + 14 = (x + 2)(x + 7)
• x² − bx + c; Signs of the two numbers both negative; Example x² − 9x + 20 = (x − 4)(x − 5)
• x² ± bx − c; Signs of the two numbers one positive, one negative; Example x² − 2x − 15 = (x − 5)(x + 3)

Worked example: x² − 2x − 15

Multiply to −15 and add to −2. The numbers have different signs.
Try −5 and +3: (−5) × 3 = −15 and −5 + 3 = −2 ✓

Answer: (x − 5)(x + 3)

Worked example: x² − 9x + 20

Multiply to +20 and add to −9: both negative. −4 × −5 = 20 and −4 + −5 = −9.

Answer: (x − 4)(x − 5)

Difference of two squares and perfect squares

x² − 9 has no x term. Think of it as x² + 0x − 9: the numbers +3 and −3 multiply to −9 and add to 0.

Key fact: a² − b² = (a + b)(a − b), for example x² − 49 = (x + 7)(x − 7).

Worked example: special cases

y² − 100 = (y + 10)(y − 10)
x² + 10x + 25: 5 × 5 = 25 and 5 + 5 = 10, so it is (x + 5)(x + 5) = (x + 5)², a perfect square.

Answer: (y + 10)(y − 10) and (x + 5)²

Common mistakes

• Choosing numbers that multiply correctly but add to the wrong value: x² + x − 12 is (x + 4)(x − 3), not (x − 4)(x + 3).
• Not taking out the highest common factor: 3x² − 12x = 3(x² − 4x) is not fully factorised.
• Trying to factorise x² + 9 as a difference of two squares. It has a plus sign, so it does not factorise.
• Not checking. Expanding your answer takes seconds and catches sign errors.

Revision questions

Try these, then check your answers below each one.

Question 1. Factorise x² + 9x + 20. (Choose from: (x + 4)(x + 5), (x + 2)(x + 10), (x + 9)(x + 20), (x - 4)(x - 5).)

Answer: (x + 4)(x + 5)

Working: 4 × 5 = 20 and 4 + 5 = 9.

Question 2. Factorise x² - 2x - 24. (Choose from: (x - 6)(x + 4), (x + 6)(x - 4), (x - 8)(x + 3), (x - 12)(x + 2).)

Answer: (x - 6)(x + 4)

Working: -6 × 4 = -24 and -6 + 4 = -2.

Question 3. Factorise fully 6x² - 9x. (Choose from: 3x(2x - 3), 3(2x² - 3x), x(6x - 9), 6x(x - 9).)

Answer: 3x(2x - 3)

Working: The HCF of 6x² and 9x is 3x: 3x(2x - 3).

Related lessons

• Previous lesson: 16.3 Using Quadratic Graphs
• Next lesson: 16.5 Solving Quadratic Equations Algebraically
• In this chapter: 16.1 Expanding Double Brackets
• In this chapter: 16.2 Plotting Quadratic Graphs

Continue in the full course

This lesson is section 16.4 (EGCMF1604) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.