How to Do Factorisation in GCSE Maths Step by Step

How to Do Factorisation in GCSE Maths Step by Step

Factorisation might seem tricky at first, but it’s actually just finding the “ingredients” that were multiplied together. Once you know the patterns, it becomes straightforward and will boost your confidence across many GCSE topics.

This guide shows you exactly how to tackle every type of factorisation with simple, clear steps you can follow.

What is Factorisation?

Factorisation is the reverse of expanding brackets. Instead of multiplying out (x + 2)(x + 3) to get x² + 5x + 6, you work backwards from x² + 5x + 6 to find (x + 2)(x + 3).

Why it matters for GCSE:

• Makes solving quadratic equations much faster
• Essential for simplifying algebraic fractions
• Helps with graph work to find x-intercepts
• High-mark questions often need factorisation

Think of it as finding the building blocks that created the expression.

Method 1: Taking Out Common Factors (Single Brackets)

This is your starting point. Look for numbers or letters that appear in every term and “take them out.”

Simple Steps

Step 1: Find the biggest number that divides into all terms Step 2: Find the highest power of each letter in every term
Step 3: Put these outside the bracket Step 4: Work out what goes inside

Worked Example 1

Factorise 12x + 8

Step 1: Biggest number that goes into both 12 and 8 both divide by 4

Step 2: No letters appear in every term

Step 3: Put 4 outside 4(…)

Step 4: Work out the inside 12x ÷ 4 = 3x 8 ÷ 4 = 2

Answer: 4(3x + 2)

Check: 4 × 3x + 4 × 2 = 12x + 8 ✓

Worked Example 2

Factorise 15x²y + 10xy²

Step 1: Biggest number 15 and 10 both divide by 5

Step 2: Letters in every term x appears as x² and x, so take out x y appears as y and y², so take out y

Step 3: Put 5xy outside 5xy(…)

Step 4: Work out inside 15x²y ÷ 5xy = 3x
10xy² ÷ 5xy = 2y

Answer: 5xy(3x + 2y)

Check: 5xy × 3x + 5xy × 2y = 15x²y + 10xy² ✓

Method 2: Factorising Quadratics (Double Brackets)

For expressions like x² + 5x + 6, you need two brackets. Find two numbers that multiply to give the last number and add to give the middle number.

Simple Steps

Step 1: Set up two brackets: (x )(x ) Step 2: Find pairs of numbers that multiply to give the last number Step 3: Pick the pair that adds to give the middle number
Step 4: Put the signs in correctly

Worked Example 3

Factorise x² + 7x + 12

Step 1: Set up brackets (x )(x )

Step 2: Pairs that multiply to give 12 1 × 12 = 12 2 × 6 = 12
3 × 4 = 12

Step 3: Which pair adds to 7? 1 + 12 = 13 ✗ 2 + 6 = 8 ✗ 3 + 4 = 7 ✓

Step 4: Put in the signs Since we need +7 and +12, both signs are positive

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

Worked Example 4

Factorise x² + x – 12

Step 1: Set up brackets (x )(x )

Step 2: Pairs that multiply to give -12 Need one positive, one negative: 1 × (-12) = -12 2 × (-6) = -12 3 × (-4) = -12 4 × (-3) = -12

Step 3: Which pair adds to +1? 1 + (-12) = -11 ✗ 2 + (-6) = -4 ✗ 3 + (-4) = -1 ✗ 4 + (-3) = 1 ✓

Step 4: Put them in Answer: (x + 4)(x – 3)

Method 3: Difference of Two Squares

This special pattern appears when you have two perfect squares subtracted: a² – b² = (a + b)(a – b)

Quick Recognition

Look for:

• Two terms only
• Both are perfect squares
• Subtraction between them

Worked Example 5

Factorise x² – 9

Recognise the pattern: x² – 3²

Use the formula: a² – b² = (a + b)(a – b) Where a = x and b = 3

Answer: (x + 3)(x – 3)

Worked Example 6

Factorise 4x² – 25

Recognise the pattern: (2x)² – 5²

Use the formula: Where a = 2x and b = 5

Answer: (2x + 5)(2x – 5)

When a ≠ 1 (Harder Quadratics)

For expressions like 2x² + 7x + 3, where the first number isn’t 1.

Simple Approach

Step 1: Look at the first number to see possible brackets Step 2: Try different combinations Step 3: Check by expanding

Worked Example 7

Factorise 2x² + 7x + 3

Step 1: First number is 2, so brackets are: (2x )(x )

Step 2: Need numbers that multiply to give 3 Only option: 1 and 3

Step 3: Try combinations (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓

Answer: (2x + 1)(x + 3)

Common Mistakes to Avoid

Mistake 1: Not Factorising Completely

Wrong: 6x² + 9x = 3(2x² + 3x) ← Not fully factorised

Right: 6x² + 9x = 3x(2x + 3) ← Always take out the biggest factor

Mistake 2: Sign Errors

Remember:

• If last number is positive: both brackets have same sign as middle term
• If last number is negative: brackets have different signs

Example: x² – 5x + 6 Since +6 and middle term is -5, both brackets are negative: (x – 2)(x – 3)

Mistake 3: Not Checking Answers

Always expand your factorised answer to check: (x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6

If this doesn’t match your original, you made an error.

Mistake 4: Confusing Different Methods

Remember:

• x² + 9 cannot be factorised (sum of squares)
• x² – 9 = (x + 3)(x – 3) (difference of squares)

When to Use Each Method

Use single brackets when:

• All terms have common factors
• Simplifying before other methods

Use double brackets when:

• You have a quadratic expression (x² term)
• Solving quadratic equations

Use difference of squares when:

• Two perfect squares subtracted
• Expressions like x² – 16 or 9x² – 4

Practice Questions

Single Brackets:

1. 8x + 12
2. 15y² – 10y
3. 6ab + 9a²b

Double Brackets: 4. x² + 8x + 15 5. x² – 3x – 10 6. x² – 6x + 9

Difference of Squares: 7. x² – 16 8. 9x² – 4 9. 25a² – 49

Solutions:

1. 4(2x + 3)
2. 5y(3y – 2)
3. 3ab(2 + 3a)
4. (x + 3)(x + 5)

(Try the rest yourself!)

Building Exam Success with Factorisation

Time-Saving Exam Tips

Quick recognition checks:

• Scan for common factors first (saves time later)
• Look for difference of squares patterns
• Check if quadratics have “nice” integer factors

In calculator papers:

• Use systematic factor-finding for large numbers
• Double-check answers by expanding
• Don’t waste time on expressions that don’t factorise

Common Exam Question Types

Direct factorisation: “Factorise completely: 6x² – 9x” Equation solving: “Solve x² + 5x + 6 = 0 by factorisation”
Simplification: “Simplify: (x² – 9)/(x + 3)” Applied problems: Setting up and solving using factorised forms

Ready to Master Factorisation?

Factorisation becomes much more manageable when you understand the patterns and have systematic approaches for each type. The key is building confidence through practice and having support when you encounter difficulties.

If factorisation feels overwhelming, you’re in good company. Many successful GCSE students initially struggle with algebraic manipulation and pattern recognition. The difference is getting the right guidance to build these essential skills.

How We Can Help

Our GCSE Mathematics tutors specialise in:

• Pattern recognition training to identify factorisation types quickly
• Systematic checking methods to eliminate algebraic errors
• Confidence building through structured progression
• Exam technique development for maximum efficiency

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