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Expand And Simplify

Here we will learn how to expand and simplify algebraic expressions. First we expand the brackets, then we collect the like terms to simplify the expression.

At the end you’ll find expanding brackets worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What does ‘expand and simplify’ mean?

In order to expand and simplify an expression, we need to multiply out the brackets and then simplify the resulting expression by collecting the like terms.

Expanding brackets (or multiplying out) is the process by which we remove brackets.

It is the reverse process of factorisation. To expand brackets we multiply everything outside of the bracket, by everything inside of the bracket.

Once we have expanded the brackets we can simplify the expression by collecting the like terms.

E.g.

If we expand and simplify

2(x+5)+3(x-2)

we will get

\begin{aligned} 2(x+5)+3(x-2) &=2 x+10+3 x-6 \\\\ &=5 x+4 \end{aligned}

What does expand and simplify mean?

What does expand and simplify mean?

Expand and simplify worksheets

Expand and simplify worksheets

Expand and simplify worksheets

Download free expand and simplify worksheets with 20+ reasoning and applied questions, answers and mark scheme.

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Expand and simplify worksheets

Expand and simplify worksheets

Expand and simplify worksheets

Download free expand and simplify worksheets with 20+ reasoning and applied questions, answers and mark scheme.

DOWNLOAD FREE

This lesson is part of our series of lessons to support revision on algebraic expressions. Related step by step guides include:

Multiplying out brackets and simplify 


β€˜Multiplying out brackets’ is another term for expanding brackets. It means exactly the same thing. β€œExpand the brackets” is the same as β€œmultiply out the brackets”, it just gives the additional clue that when we expand brackets, we are multiplying everything outside the brackets by everything inside the brackets.

How to expand and simplify brackets

In order to expand and simplify brackets:

  1. Expand each bracket in the expression.
  2. Collect the like terms.

There are three ways to expand and simplify brackets as covered below:

  • single brackets
  • two or more brackets
  • surds

Explain how to expand and simplify brackets?

Explain how to expand and simplify brackets?
  1. Expand and simplify with single brackets.

Expand the brackets to give the following expression:

E.g. 2(x + 5) + 3(x βˆ’ 1)
= 2x + 10 + 3x βˆ’ 3)
= 5x + 7

Remember: expressions with two terms like 5x + 7 are known as binomials.

2Expand and simplify with two or more brackets.

Expand the brackets to give the following expression:

E.g. (x + 5)(x βˆ’ 1) 
= x2 + 5x βˆ’ x βˆ’ 5
= x2 + 4x βˆ’ 5

Remember: expressions with three terms like x2 + 4x βˆ’ 5 are known as trinomials.

An expression that contains more than two terms and includes variables and coefficients is called a polynomial.

3 Expand and simplify with surds.

E.g. (3 + √5)(2 + √5) 
= 6 + 3√5 + 2√5 + √5√5
= 11 + 5√5

1) Expand and simplify with single brackets

To expand a single bracket we multiply the term outside of the bracket by everything inside of the bracket. We can simplify the expression by collecting the like terms.

Expand and simplify examples (with single brackets)

Example 1: constants outside of the brackets

Expand and simplify:

2(x + 5) + 3(x βˆ’ 2)

  1. Expand each bracket in the expression

Multiply the first bracket:

βœ•x+ 5
22x+ 10

Multiply the second bracket – remember we are multiplying both x and βˆ’ 2 by + 3:

βœ•xβˆ’ 2
+ 3+ 3xβˆ’ 6

Remember to include the – sign in front of the number.

2(x + 5) + 3(x βˆ’ 2)

 = 2x + 10 + 3x βˆ’ 6

2Collect the like terms

Highlight the two x terms (2x and + 3x) and the two constants (+ 10 and βˆ’ 6).

Remember to highlight the sign in front of the number too!

2x + 10 + 3x βˆ’ 6

= 5x + 4

Example 2: constants and variables outside of the brackets

Expand and simplify:

2x(x + 6) - 3(x - 2)

Expand each bracket in the expression.

Collect the like terms.

Example 3: variables in both terms in the brackets

Expand and simplify:

3(2x βˆ’ 6y) βˆ’ 5(x βˆ’ 2y)

Expand each bracket in the expression.

Collect the like terms.

Practice expand and simplify questions (with single brackets)

1. Expand and Simplify: 3(x+7)-2(x+3)

5x+15
GCSE Quiz False

x+15
GCSE Quiz True

x+27
GCSE Quiz False

5x+27
GCSE Quiz False
3(x+7)-2(x+3)

 

Expand each bracket

 

3x+21-2x-6

 

Collect like terms

 

x+15

2. Expand and simplify: 8(y-5)-5(y-2)

13y-30
GCSE Quiz False

3y-50
GCSE Quiz False

3y-30
GCSE Quiz True

13y-50
GCSE Quiz False
8(y-5)-5(y-2)

 

Expand each bracket

 

8y-40-5y+10

 

Collect like terms

 

3y-30

3. Expand and simplify: 5x(3x-2)-4x(2x+3)

23x^{2}-22x
GCSE Quiz False

7x^{2}-5
GCSE Quiz False

7x^{2}-22x
GCSE Quiz True

7x^{2}-2x
GCSE Quiz False
5x(3x-2)-4x(2x+3)

 

Expand each bracket

 

15x^{2}-10x-8x^{2}-12x

 

Collect like terms

 

7x^{2}-22x

4. Expand and simplify: 5(6x-2y)-2(8x-5y)

14x
GCSE Quiz True

14x-20y
GCSE Quiz False

46x
GCSE Quiz False

14x+20y
GCSE Quiz False
5(6x-2y)-2(8x-5y)

 

Expand each bracket

 

30x-10y-16x+10y

 

Collect like terms

 

14x

2) Expand and simplify with two or more brackets

To expand two or more brackets we multiply every term in the first bracket by every term in each of the other brackets.

Expand and simplify examples (with two or more brackets)

Example 1: variables have a coefficient of 1

Expand and simplify:

(x + 5)(x βˆ’ 1)

  1. Expand the brackets in the expression.

βœ•xβˆ’ 1
xx2βˆ’ x
+ 5+ 5xβˆ’ 5

x βœ• x = x2
x βœ• βˆ’ 1 = βˆ’ x

+ βœ• βˆ’ = βˆ’ so the answer is negative.

x βœ• 5 = 5x 
5 βœ• βˆ’ 1 = βˆ’ 5

+ βœ• βˆ’ = βˆ’ so the answer is negative.

(x + 5)(x βˆ’ 1) = x2 βˆ’ x + 5x βˆ’ 5

2Collect the like terms.

The only like terms we have are the two x terms (βˆ’ x and + 5x). Highlight them both.

x2 βˆ’ x + 5x βˆ’ 5

= x2 + 4x βˆ’ 5

Example 2: variables have a coefficient of greater than 1

Expand and simplify:

(2x βˆ’ 4)(x + 5)

Expand the brackets in the expression.

Collect the like terms.

Example 3: with triple brackets

Expand and simplify:

(x + 3)2(x βˆ’ 1)

Expand and simplify the first two brackets in the expression.

Multiply this new expression with the third bracket and then simplify by collecting like terms.

Practice expand and simplify questions (with two or more brackets)

1. Expand and simplify: (x+3)(x-7)

x^{2}-4x+21
GCSE Quiz False

x^{2}-10x-21
GCSE Quiz False

x^{2}-4x-21
GCSE Quiz True

x^{2}-10x+21
GCSE Quiz False
(x+3)(x-7)

 

Expand the brackets

 

x^{2}-7x+3x-21

 

and collect like terms

 

x^{2}-4x-21

2. Expand and simplify: (2x-4)(x-9)

2x^{2}-14x+36
GCSE Quiz False

2x^{2}-14x-36
GCSE Quiz False

2x^{2}-22x+36
GCSE Quiz True

2x^{2}-22x-36
GCSE Quiz False
(2x-4)(x-9)

 

Expand the brackets

 

2x^{2}-18x-4x+36

 

and collect like terms

 

2x^{2}-22x+36

3. Expand and simplify: (x-3)^2(2x+1)

2x^{3}-13x^{2}+12x+9
GCSE Quiz False

2x^{3}+13x^{2}+12x+9
GCSE Quiz False

2x^{3}-11x^{2}+12x+9
GCSE Quiz True

2x^{3}-11x^{2}-24x+9
GCSE Quiz False
(x-3)^{2}(2x+1)

 

can be written as
(x-3)(x-3)(2x+1)

 

Expanding the first two brackets gives

 

(x^{2}-3x-3x+9)(2x+1)

 

(x^{2}-6x+9)(2x+1)

 

then expanding again

 

2x^{3}+x^{2}-12x^{2}-6x+18x+9

 

and collecting like terms

 

2x^{3}-11x^{2}+12x+9

4. Expand and simplify: (2x-1)^{3}

8x^{3}+8x^{2}+6x+1
GCSE Quiz False

8x^{3}+12x^{2}+12x+1
GCSE Quiz False

8x^{3}-12x^{2}+6x-1
GCSE Quiz True

2x^{3}+12x^{2}+6x+1
GCSE Quiz False
(2x-1)^{3}

 

can be written as
(2x-1)(2x-1)(2x-1)

 

Expanding two of the brackets gives

 

(2x-1)(4x^{2}-2x-2x+1)

 

(2x-1)(4x^{2}-4x+1)

 

and expanding again

 

8x^{3}-8x^{2}+2x-4x^{2}+4x-1

 

then collect like terms

 

8x^{3}-12x^{2}+6x-1

3) Expand and simplify with surds

To expand the brackets we need to multiply each term by every other term.

Expand and simplify examples (with surds)

Example 1: with constants and surds

Expand and simplify:

(3 + √5)(2 + √5) 

  1. Expand the brackets in the expression.

βœ•2+ √5
36+ 3√5
+ √5+ 2√5+ 5

3 βœ• 2 = 6
3 βœ• √5 = 3√5 
2 βœ• √5 = 2√5
√5 βœ• √5 = 5 

= 6 + 3√5 + 2√5 + 5

2Collect the like terms.

Highlight the two constant terms (6 and 25) and highlight the two surd terms (3√5 and 2√5).

6 + 3√5 + 2√5 + 25

= 11 + 5√5

Example 2: with all terms as surds

Expand and simplify:

(√2 + √5)2 βˆ’ (3 + √5)2 

Expand the brackets in the expression.

Collect the like terms.

Practice expand and simplify questions (with surds)

1. Expand and simplify: (2+\sqrt{6})(3+\sqrt{6})\\

32+5 \sqrt{6}
GCSE Quiz False

12+5\sqrt{6}
GCSE Quiz True

12+6\sqrt{6}
GCSE Quiz False

7\sqrt{6}
GCSE Quiz False

(2+\sqrt{6})(3+\sqrt{6})\\
=6+2\sqrt{6}+3\sqrt{6}+6\\
=12+5\sqrt{6}

2. Expand and simplify: (3+\sqrt{5})(2-\sqrt{5})\\

11- \sqrt{5}
GCSE Quiz False

11-5\sqrt{5}
GCSE Quiz False

1-5\sqrt{5}
GCSE Quiz False

1-\sqrt{5}
GCSE Quiz True

(3+\sqrt{5})(2-\sqrt{5})\\
=6-3\sqrt{5}+2\sqrt{5}-5\\
=1-5\sqrt{5}

3. Expand and simplify: (\sqrt{3}+\sqrt{7})^{2}-(3+\sqrt{7})\\

13+2\sqrt{21}-\sqrt{7}
GCSE Quiz False

7+2\sqrt{21}+\sqrt{7}
GCSE Quiz False

7+\sqrt{21}-\sqrt{7}
GCSE Quiz False

7+2\sqrt{21}-\sqrt{7}
GCSE Quiz True

(\sqrt{3}+\sqrt{7})^{2}-(3+\sqrt{7})\\
=\sqrt{21}+\sqrt{21}+3+7-3-\sqrt{7}\\
=7+2\sqrt{21}-\sqrt{7}

4. Expand and simplify: (\sqrt{2}-\sqrt{8})^{2}-(\sqrt{2}+\sqrt{8})^{2}\\

16\sqrt{2}
GCSE Quiz False

-16
GCSE Quiz True

16\sqrt{8}
GCSE Quiz False

16
GCSE Quiz False

(\sqrt{2}-\sqrt{8})^{2}-(\sqrt{2}+\sqrt{8})^{2}\\
=2-2\sqrt{16}+8-2-2\sqrt{16}-8\\
=-16

Common misconceptions

  • Multiplying all terms in the bracket

We must multiply the value outside the brackets by every term inside the brackets (parentheses).

E.g.

2(6x2  βˆ’ 3x) = 12x2 βˆ’ 3x βœ–

Here we have not multiplied the value outside of the brackets by the second term.

The correct answer is:

2(6x2 βˆ’ 3x) = 12x2 βˆ’ 6x βœ”

  • Multiplying with negative numbers

For two numbers to multiply to give a + their signs must be the same.

For two numbers to multiply to give a βˆ’ their signs must be the different.

E.g.

βˆ’ 4(3y βˆ’ 5) = βˆ’ 4y βˆ’ 20

Here we have not used βˆ’ βœ• βˆ’ = +

βˆ’ 4 βœ• βˆ’ 5 = + 20

So the correct answer is βˆ’ 4(3y βˆ’ 5) = βˆ’ 4y + 20.

  • Squaring a term

When we square something, we multiply it by itself.

E.g.

32 = 3 βœ• 3
x2 = x βœ• x
(5y)2 = 5y βœ• 5y

When we square a bracket, we multiply it by the entire bracket.

(x + 3)2 = (x + 3)(x + 3)
= x2 + 6x + 9
NOT x2 + 9

  • Collecting like terms

When we collect like terms we must include the sign in front of the number.

E.g.

3x + 8 + 6x βˆ’ 2
NOT 3x + 8 + 6x βˆ’ 2

Expand and simplify GCSE questions

Learning checklist

  • Multiply a single term over a bracket
  • Expand products of 2 or more binomials
  • Simplify and manipulate algebraic expressions by collecting like terms

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