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Factorise the Difference of Two Squares (DOTS)

In this worksheet, students will factorise the difference of two squares.

'Factorise the Difference of Two Squares (DOTS)' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Look at these expressions:

 

y² - 36

x² - 121

49 - p²

 

These are called the difference of two squares. Both terms in these expressions are squared: y² and 36, x² and 121, 49 and p².

You should look out for square numbers (1, 4, 9, 16, 25, ...) and a subtraction sign (this is what 'difference' means).

 

To factorise, set up two brackets, one with a minus and one with a plus sign and square root the terms.

y² - 36 = (y - 6)(y + 6)

x² - 121 = (x - 11)(x + 11)

49 - 4p² = (7 - 2p)(7 + 2p)

Tick the expressions which are the difference of two squares.

y² - 81

154 - a²

t² + 9

x² - 144

Match each expression to the correct factorisation.

Column A

Column B

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

Choose which of the statements are true and which are false.

 TrueFalse
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 the expressions which are a difference of two squares.

x² + 9

8x² - 49

4x² - 81

x² - 144

Write 100g² - 49 as a difference of two squares.

Write 0.36r² - 0.04 as a difference of two squares.

Match the expressions to their correct factorisation as a difference of two squares.

Column A

Column B

v¹4; - p¹2;
(12v³ - 0.3)(12v³ + 0.3)
144v&sup6; - 0.09
(v&sup4; - 2p)(v&sup4; + 2p)
v&sup8; - 4p²
(v&sup7; - p&sup6;)(v&sup7; + p&sup6;)
81v&sup9; - 0.16
(9v³ - 0.4)(9v³ + 0.4)

Tick the correct factorisations of these differences of two squares.

Factorise completely 4x2 - 9y2.

Factorise 9x² - 25y².

  • Question 1

Tick the expressions which are the difference of two squares.

CORRECT ANSWER
y² - 81
x² - 144
EDDIE SAYS
Only the first and the last expressions are what we call the difference of two squares: both terms are square and there is a minus sign between them.
  • Question 2

Match each expression to the correct factorisation.

CORRECT ANSWER

Column A

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 to set up two brackets with different signs (a minus and a plus) and square root both terms of the expressions.
  • Question 3

Choose which of the statements are true and which are false.

CORRECT ANSWER
 TrueFalse
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 minus between them. To factorise you need to square root each term and write them in two brackets, one with a plus sign, one with a minus sign.
  • Question 4

Tick the expressions which are a difference of two squares.

CORRECT ANSWER
4x² - 81
x² - 144
EDDIE SAYS
A difference means subtractions, so we need an expression with a minus (so the first one is definitely wrong. Out of the other three, we need to pick an expression with both terms being square numbers. In option 2 8x² is not a square term, even though 49 is, so this expression is not a difference of two squares. The remaining two expressions are made up of two square terms, so they are a difference of two squares.
  • Question 5

Write 100g² - 49 as a difference of two squares.

CORRECT ANSWER
(10g-7)(10g+7)
(10g - 7)(10g + 7)
(10g- 7)(10g+ 7)
(10g -7)(10g +7)
EDDIE SAYS
Remember to square root each term. This will give us 10g and 7. Now put these in two brackets, one with a minus, one with a plus: (10g - 7)(10g + 7)
  • Question 6

Write 0.36r² - 0.04 as a difference of two squares.

CORRECT ANSWER
(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. The correct answer is (0.6r - 0.2)(0.6r + 0.2).
  • Question 7

Match the expressions to their correct factorisation as a difference of two squares.

CORRECT ANSWER

Column A

Column B

v¹4; - p¹2;
(v&sup7; - p&sup6;)(v&sup7; + p&s...
144v&sup6; - 0.09
(12v³ - 0.3)(12v³ + 0.3...
v&sup8; - 4p²
(v&sup4; - 2p)(v&sup4; + 2p)
81v&sup9; - 0.16
(9v³ - 0.4)(9v³ + 0.4)
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). Can you see how each of the powers was reduced when factorising? 14 became 7, 6 became 3 and so on.
  • Question 8

Tick the correct factorisations of these differences of two squares.

CORRECT ANSWER
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 brackets.
  • Question 9

Factorise completely 4x2 - 9y2.

CORRECT ANSWER
(2x - 3y)(2x + 3y)
(2x-3y)(2x+3y)
(2x- 3y)(2x+ 3y)
(2x -3y)(2x +3y)
EDDIE SAYS
Both terms of 4x² - 9y² are square terms, so we need to square root them and put these in two brackets, one with a minus, one with a plus. Square root of 4x² is 2x. Square root of 9y² is 3y. Now put these in brackets: (2x - 3y)(2x + 3y).
  • Question 10

Factorise 9x² - 25y².

CORRECT ANSWER
(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? Square root of 9x² is 3x. Square root of 25y² 5y. Now put these in double brackets with a plus in one and a minus in the other bracket: (3x - 5y)(3x + 5y).
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