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Weekly online one to one GCSE maths revision lessons available in the spring term

In order to access this I need to be confident with:

Expanding brackets

Factors and multiples

Terms, Expressions, Equations, Formulas

Terms, Expressions, Equations, Formulas

Multiplying Negative Numbers

This topic is relevant for:

factorising quadratics in the form

We have previously learnt about factorising expressions into a single bracket and factorising quadratics into double brackets.

Here we will learn how to factorise a quadratic in the form ^{2} – b^{2}

(Remember: ^{2} – b^{2}

There are also factorising worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

When we have a squared term subtracted from another squared term we can use the difference of two squares method. Square numbers are sometimes called perfect squares.

To use quadratic factorisation on an expression in the form ^{2} – b^{2}

a^{2} – b^{2} = (a + b)(a – b)

In order to factorise an algebraic expression using the difference of two squares:

- Write down two brackets
- Square root the first term and write it on the left hand side of both brackets
- Square root the last term and write it on the right hand side of both brackets
- Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)

**Example 1**

Fully factorise

^{2} – 9

- Write down two brackets

2Square root the first term and write it on the left hand side of both brackets

^{2} – 9

^{2}

3Square root the last term and write it on the right hand side of both brackets.

^{2} – 9)

4Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter).

We can check the final answer by multiplying out the brackets!

^{2} – 9

**Example 2**

Fully factorise

^{2}

Write down two brackets

Square root the first term and write it on the left hand side of both brackets

^{2}

Square root the last term and write it on the right hand side of both brackets

^{2}

^{2}

Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)

We can check the final answer by multiplying out the brackets!

^{2}

**Example 3**

Fully factorise

^{2} – 16

Write down two brackets

Square root the first term and write it on the left hand side of both brackets

^{2} – 16

^{2}

Square root the last term and write it on the right hand side of both brackets

^{2} – 16

Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.

We can check the final answer by multiplying out the brackets!

^{2} – 16

**Example 4**

Fully factorise

^{2} – 81y^{2}

Write down two brackets

Square root the first term and write it on the left hand side of both brackets

^{2} – 81y^{2}

^{2}

Square root the last term and write it on the right hand side of both brackets

^{2} – 81y^{2}

^{2}

Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)

We can check the final answer by multiplying out the brackets!

^{2} – 81y^{2}

**Example 5**

Fully factorise

^{3} – 64x

Be careful, this one is not the difference of two squares!

We first need to find the greatest common factor (

^{2} – 64)

Write down two brackets with the x at the front

Square root the first term and write it on the left hand side of both brackets

^{2} – 64)

^{2}

Square root the last term and write it on the right hand side of both brackets

^{2} – 64)

Put a + in the middle of one bracket and a – in the middle of the other (the order doesn’t matter.)

We can check the final answer by multiplying out the brackets!

^{3} – 64x

- There must be a + in one bracket, and a – in the other, the order doesn’t matter.
( + )( – ) or( – )( + ) - Remember to square root the entire term including the coefficients of the variables.

E.g.9x ^{2}– 49√9x ^{2}= 3x√49 = 7 - The term factorising can sometimes be written as ‘factoring’ or factorization’

1. Fully factorise x^{2} – 25

Show answer

= (x + 5)(x – 5)

2. Fully factorise: y^{2} – 81

Show answer

= (y + 9)(y – 9)

3. Fully factorise: 49 – y^{2}

Show answer

= (7 + y)(7 – y)

4. Fully factorise: 4 – x^{2}

Show answer

= (2 + x)(2 – x)

5. Fully factorise: 16x^{2} – 100

Show answer

= (4x + 10)(4x -10)

6. Fully factorise: 49 – 9y^{2}

Show answer

= (7 + 3y)(7 – 3y)

7. Fully factorise: 81x^{2} – 16y^{2}

Show answer

= (9x + 4y)(9x – 4y)

8. Fully factorise: 4x^{3} – 36x

Show answer

= 4x(x + 3)(x – 3)

1. Factorise x^{2} – 100

Show answer

= (x + 10)(x – 10)

(2 marks)

2. Factorise y^{2} – 49

Show answer

= (y + 7)(y – 7)

(2 marks)

3. Factorise 2x^{2} – 50

Show answer

= 2(x^{2} – 25) = 2(x + 5)(x – 5)

(2 marks)

Download for free one or more factorising worksheets to help you practise this further and more.

You have now learnt to:

- Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
- Factorise quadratic expressions of the form x
^{2}+ bx + c - Factorise quadratic expressions of the form of the difference of two squares.
- Factorising quadratic expressions of the form ax
^{2}+ bx + c (H)

Head back to the main Factorising page for a recap on all types of factorising, along with, mixed practice questions, mixed worksheets and exam questions.

The next topics are:

- Solving Equations
- Solving Linear Equations
- Solving Quadratic Equations
- Quadratic Formula
- Completing the square
- Forming and solving quadratic equations

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