 # Factorise the Difference of Two Squares (DOTS)

In this worksheet, students will practise identifying expressions which use DOTS, and then factorise or expand pairs of brackets to create alternative expressions of the same value. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Expressions

Difficulty level:   ### QUESTION 1 of 10

Look at these expressions:

y² - 36

x² - 121

49 - p²

These are all examples of the difference of two squares (DOTS).

Both terms in these expressions are squares:  and 36 and 121, and 49.

To spot DOTS, we need to look out for square numbers (1, 4, 9, 16, 25, etc.) and a subtraction sign (this is what 'difference' means in the title).

To factorise DOTS expressions, set up two brackets (one with a minus and one with a plus sign) and then square root the terms within each:

e.g. y² - 36 = (y - 6)(y + 6)

e.g. x² - 121 = (x - 11)(x + 11)

49 - 4p² = (7 - 2p)(7 + 2p)

Can you see how the numbers can be multiplied together to become the square number in the original expressions?

Try working back through each example above by expanding the brackets back out to check you reach the original expression.

It works, doesn't it?!

In this activity, we will practise identifying expressions which use DOTS and then factorise or expand pairs of brackets to create alternative expressions of the same value.

Tick all the expressions below which are examples of the difference of two squares (DOTS).

y2 - 81

154 - a2

t2 + 9

x2 - 144

Match each expression below to its correct factorisation.

## Column B

x2 - 25
(x - 5)(x + 5)
x2 - 4
(x - 2)(x + 2)
x2 - 100
(x - 10)(x + 10)
x2 - 225
(x - 15)(x + 15)

Choose whether each of the statements below are true or false.

 True False x² - 36y² = (x + 6y)(x - 6y) 6x² - 9 = (3x - 3)(3x + 3) 36y² - 1 = (6y - 1)(6y + 1) x² + 25 = (x + 5)(x - 5)

Tick all the expressions below which are examples of the difference of two squares (DOTS).

x2 + 9

8x2 - 49

4x2 - 81

x2 - 144

Write the expression below as a difference of two squares:

100g² - 49

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

Write the expression below as a difference of two squares:

0.36r² - 0.04

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

Match each expression below to its correct factorisation as a difference of two squares.

## Column B

v14 - p12
(9v3 - 0.4)(9v3 + 0.4)
144v6 - 0.09
(12v3 - 0.3)(12v3 + 0.3)
v8 - 4p2
(v4 - 2p)(v4 + 2p)
81v9 - 0.16
(v7 - p6)(v7 + p<...

Tick the correct factorisations of these differences of two squares.

Factorise completely:

4x<sup>2</sup> - 9y<sup>2</sup>

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

Factorise completely:

9x² - 25y²

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

• Question 1

Tick all the expressions below which are examples of the difference of two squares (DOTS).

y2 - 81
x2 - 144
EDDIE SAYS
In order to be an example of DOTS, both terms in the expression must be squares and there must be a subtraction sign. The first, third and fourth expressions include two squares: y2 and 81; t2 and 9; x2 and 144. However, the second expression uses a + sign rather than a -, so it is not an example of DOTS as a result. Look out for your square numbers and a subtraction sign to be able to spot these quickly.
• Question 2

Match each expression below to its correct factorisation.

## Column B

x2 - 25
(x - 5)(x + 5)
x2 - 4
(x - 2)(x + 2)
x2 - 100
(x - 10)(x + 10)
x2 - 225
(x - 15)(x + 15)
EDDIE SAYS
Remember our steps here: 1) Set up two brackets with different signs (a minus and a plus); 2) Square root both terms of the expression. If we follow these two simple rules then: x2 - 25 = (√x2 - √25)(√x2 + √25) = (x - 5)(x + 5) Does that make sense?
• Question 3

Choose whether each of the statements below are true or false.

 True False x² - 36y² = (x + 6y)(x - 6y) 6x² - 9 = (3x - 3)(3x + 3) 36y² - 1 = (6y - 1)(6y + 1) x² + 25 = (x + 5)(x - 5)
EDDIE SAYS
The difference of two squares must have two square terms and a subtraction sign between them. To factorise you need to square root each term and write them in two brackets, one with a plus sign and one with a minus sign. e.g. x² - 36y² = = (√ x² - √36y²) = (x + 6y)(x - 6y) Which options do not match when you have worked them out in this way? These ones are 'false'.
• Question 4

Tick all the expressions below which are examples of the difference of two squares (DOTS).

4x2 - 81
x2 - 144
EDDIE SAYS
'Difference' means subtraction, so we need to check that all the expressions have a minus sign. This eliminates the first option immediately, as it has a + sign rather than a -. Out of the other three, we need to tick expressions where both terms are squares. In option 2, 8x² is not a square term, even though 49 is, so this expression is not an example of DOTS. The remaining two expressions are made up of two square terms with a minus sign, so they are both examples of DOTS.
• Question 5

Write the expression below as a difference of two squares:

100g² - 49

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

(10g-7)(10g+7)
(10g - 7)(10g + 7)
(10g- 7)(10g+ 7)
(10g -7)(10g +7)
EDDIE SAYS
Remember our steps here: 1) Set up two brackets with different signs (a minus and a plus); 2) Square root both terms of the expression. If we follow these two simple rules then: 100g2 - 49 = (√100g2 - √49)(√100g2 + √49) = (10g - 7)(10g + 7)
• Question 6

Write the expression below as a difference of two squares:

0.36r² - 0.04

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

(0.6r-0.2)(0.6r+0.2)
(0.6r -0.2)(0.6r +0.2)
(0.6r- 0.2)(0.6r+ 0.2)
(0.6r - 0.2)(0.6r + 0.2)
EDDIE SAYS
Did you remember to square root the terms? Decimals can also be square rooted, so don't be put off by this question. √0.36 = 0.6 √0.04 = 0.2 So the correct answer is: (0.6r - 0.2)(0.6r + 0.2)
• Question 7

Match each expression below to its correct factorisation as a difference of two squares.

## Column B

v14 - p12
(v7 - p6)(v...
144v6 - 0.09
(12v3 - 0.3)(12v3...
v8 - 4p2
(v4 - 2p)(v4
81v9 - 0.16
(9v3 - 0.4)(9v3
EDDIE SAYS
We won't always remove the powers altogether from the brackets. You might need to square root the powers as well, which is equivalent to dividing them by 2. Let's look at an example together: 144v6 - 0.09 = (√144v6 - √0.09)(√144v6 + √0.09) = (12v3 - 0.3)(12v3 + 0.3) Can you see that to square root the power (6), we just divide it by two (6 ÷ 2 = 3)? Give the others a try on your own now.
• Question 8

Tick the correct factorisations of these differences of two squares.

EDDIE SAYS
You could approach this task in two ways. You could factorise the expressions at the top. Remember to square root each term; this will help you find the appropriate brackets. You could also expand and simplify the brackets on the side. This would let you work out which of the expressions, match the bracketed options. Which do you prefer? Let's do one together in both ways. 1) x² - 4y² = (√x² - √4y²)(√x² + √4y²) = (x - 2y)(x + 2y) 2) (x - 2y)(x + 2y) expanded using FOIL: x × x = x2 x × 2y = 2xy x × -2y = -2xy -2y × 2y = -4y2 The 2xy and -2xy cancel each other out, so our answer is: x2 - 4y2 Can you work out the other matches independently using your preferred method and this example to help you?
• Question 9

Factorise completely:

4x<sup>2</sup> - 9y<sup>2</sup>

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

(2x-3y)(2x+3y)
(2x - 3y)(2x + 3y)
(2x- 3y)(2x+ 3y)
(2x -3y)(2x +3y)
EDDIE SAYS
Both terms of 4x² - 9y² are squares and we have a minus sign - great news! So we need to square root them and put these in two brackets, one with a minus and one with a plus. √4x² = 2x √9y² = 3y Now let's put these into our brackets with the correct signs: (2x - 3y)(2x + 3y)
• Question 10

Factorise completely:

9x² - 25y²

Do not use a space between your brackets or the terms within your brackets or you may be marked wrong.

(3x-5y)(3x+5y)
(3x - 5y)(3x + 5y)
(3x- 5y)(3x+ 5y)
(3x -5y)(3x +5y)
EDDIE SAYS
Did you remember to square root each of the terms? √9x² = 3x √25y² = 5y Now let's put these in double brackets with a + in one and a - in the other bracket: (3x - 5y)(3x + 5y) Great work using DOTS to factorise in this activity! Hopefully you kept your eyes open for those common square numbers - why not revise these now if you found these tricky to spot?
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