Factorising quadratics Revision Notes for Edexcel GCSE Maths

Factorising quadratics

What is factorising quadratics?

Factorising quadratics means writing a quadratic expression as the product of two brackets. This is the reverse process of expanding brackets, so you need to be confident with expanding double brackets before attempting factorisation.

For example: $(x + 2)(x + 3) = x^2 + 5x + 6$ , so $x^2 + 5x + 6 = (x + 2)(x + 3)$

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Factorising is essentially "undoing" the expansion of brackets. If you can expand $(x + a)(x + b)$ to get $x^2 + (a+b)x + ab$ , then factorising reverses this process.

Basic method for factorising x² + bx + c

To factorise an expression in the form $x^2 + bx + c$ , you need to find two numbers that:

Add up to give the coefficient of x (which is b)
Multiply together to give the constant term (which is c)

The process

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Step-by-step factorisation method:

Look at the quadratic expression $x^2 + bx + c$
Find two numbers whose sum equals b and whose product equals c
Write these numbers in brackets: $(x + \text{first number})(x + \text{second number})$
Always check your answer by expanding the brackets

Understanding the signs

This table helps you work out whether the numbers in your brackets will be positive or negative:

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Signs in factorisation:

Value of b	Value of c	What this means
positive	positive	Both numbers are positive
positive	negative	Larger number is positive, smaller number is negative
negative	negative	Larger number is negative, smaller number is positive
negative	positive	Both numbers are negative

Worked example: Factorise x² - x - 20

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Worked Example: Factorising $x^2 - x - 20$

Step 1: Identify b and c

$b = -1$ (coefficient of x)
$c = -20$ (constant term)

Step 2: Find factor pairs of 20 The factor pairs of 20 are: 1 and 20, 2 and 10, 4 and 5

Step 3: Check which pair works We need two numbers that add to -1 and multiply to -20.

$4 + (-5) = -1$ ✓
$4 \times (-5) = -20$ ✓

Step 4: Write the answer $x^2 - x - 20 = (x + 4)(x - 5)$

Step 5: Check by expanding $(x + 4)(x - 5) = x^2 - 5x + 4x - 20 = x^2 - x - 20$ ✓

Special case: Difference of two squares

When you have an expression in the form $a^2 - b^2$ , this is called the difference of two squares.

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Rule: $a^2 - b^2 = (a + b)(a - b)$

This works when:

There are only two terms
Both terms are perfect squares
There's a minus sign between them

Worked example: Factorise x² - 9

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Worked Example: Factorising $x^2 - 9$

Step 1: Recognise this as difference of two squares $x^2 - 9 = x^2 - 3^2$

Step 2: Apply the rule $x^2 - 3^2 = (x + 3)(x - 3)$

Step 3: Check by expanding $(x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9$ ✓

Exam tips

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Essential exam strategies:

The answer will always have two sets of brackets
If the last term is negative, the brackets will have one + sign and one - sign
Always check your answer by expanding the brackets
Look out for difference of two squares - they're often worth easy marks!

Remember!

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Key Points to Remember:

Find two numbers that add to b and multiply to c
Use the signs table to work out positive and negative combinations
Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$
Always check your answer by expanding back
Practice makes perfect - the more you do, the quicker you'll spot the factor pairs!

Factorising quadratics (Edexcel GCSE Maths): Revision Notes