Factorising Quadratics Revision Notes for AQA 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 double brackets.

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

You should be confident with expanding double brackets before attempting factorising.

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Always check your answer by expanding the brackets back out. This is the best way to verify your factorisation is correct.

Factorising x² + bx + c

When factorising expressions in the form $x^2 + bx + c$ , you need to find two numbers that:

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

The factorised form will be $(x + \text{first number})(x + \text{second number})$ .

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Worked Example: Factorising $x^2 + 7x + 10$

Step 1: Find two numbers that add to 7 and multiply to 10

Step 2: Try different combinations:

$5 + 2 = 7$ ✓
$5 \times 2 = 10$ ✓

Step 3: Write the factorised form: $x^2 + 7x + 10 = (x + 5)(x + 2)$

Using the signs table

This table helps you determine the signs of the two numbers you need to find based on the values of b and c:

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Signs Table for Factorising:

When b positive, c positive: Both numbers are positive
When b positive, c negative: Bigger number positive, smaller number negative
When b negative, c negative: Bigger number negative, smaller number positive
When b negative, c positive: Both numbers are negative

Understanding these sign patterns will help you avoid common mistakes when factorising.

Worked example: x² - x - 20

Let's work through a complete example using the systematic approach:

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

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

Step 2: Check which pair gives the correct middle term Since $b = -1$ and $c = -20$ , we need the bigger number positive and smaller number negative.

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

Step 3: Write the factorised form $x^2 - x - 20 = (x + 4)(x - 5)$

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

Difference of two squares

Difference of two squares is a special type of quadratic that follows the pattern:

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

Key features:

Only has two terms
Both terms are perfect squares
Connected by a minus sign
No middle term (coefficient of x is zero)

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The difference of two squares pattern is one of the most important algebraic identities. Once you recognise this pattern, factorising becomes much quicker.

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Worked Example: $x^2 - 9$

Since $9 = 3^2$ , we have: $x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)$

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

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Worked Example: $x^2 - 36$

Since $36 = 6^2$ , we have: $x^2 - 36 = (x + 6)(x - 6)$

Top tips for success

Here are the most effective strategies for mastering quadratic factorisation:

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Key Strategies for Factorising Success:

Try factor pairs of the constant term c when factorising $x^2 + bx + c$
Always expand your brackets to check your answer
Look for patterns - especially difference of two squares
Use the signs table to determine whether numbers should be positive or negative
Practice regularly - factorising gets easier with repetition

Remember!

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Essential Points to Remember:

Factorising means writing as a product of two brackets
Find two numbers that add to b and multiply to c
Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$
Always check your answer by expanding the brackets
Use the signs table to determine positive and negative numbers

Factorising Quadratics (AQA GCSE Maths): Revision Notes