Expanding and Factorising – GCSE Maths

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Introduction

Expanding:

  • Simplifying the expression or equation by eliminating the brackets present. Multiplying the number outside the brackets with numbers inside.

Factorising:

  • Writing the simplified equation in a brief form means using brackets.

Diagram showing the expansion of algebraic expressions 6(x + 3) = 6x + 18 and 2(y + 30) = 2y + 60 with arrows illustrating the process of Expanding and Factorising

Expansion and Factorisation in Detail

Expansion:

Example:

Expression 2(3x + 5) + 7x showing an algebraic equation before expanding and simplifying like terms.

    • Step#1: Identify the paranthesis in expression –
    • Step#2: Apply the distributive property, multiply the term outside the paranthesis by the term inside –

Expanding and Factorising: Step-by-step breakdown of 2(3x + 5) + 7x into expanded and simplified terms.

    • Step#3: Combine the terms with same variable if any –

Expanding and Factorising: Algebraic expression simplified to 13x + 10

Factorisation:

Example:

Expression showing 5x plus 10 plus 3y plus 9

    • Step#1: Find the common factors –

Algebraic expressions 5x, 10, 3y, and 9 broken down into factors showing common factors

    • Step#2: Combine the terms having common factors-

Algebraic expression written as 5(x + 2) + 3(y + 3)

Expand the following algebric equations:

  • These are also used as general formulae.

Visual showing expansion and simplification of (a + b)², (a - b)², and (a + b)(a - b)

certified Physics and Maths tutorSolved Example

Problem: Represent the multiplication of three consecutive integers in algebric form. Write the equation by expanding it further.

Solution: 

Suppose x is an integer:

Worked example showing expansion of x(x + 1)(x + 2) into a cubic expression

Final Answer: x3 + 3x2 + 2x = 0

certified Physics and Maths tutorSolved Example

Problem: What will be the total price of 2 sets of 5 toffees and 6 chocolates if the price individually is represented by variables?

Solution: 

Suppose the price of one toffees is x and that of one chocolate is y, then the total price will be:

Factorised expression 2 times the bracket of 5x plus 6y

Expansion of expression 2 times 5x plus 6y shown using sweets, toffees, and chocolates

Final Answer: 10x + 12y

certified Physics and Maths tutorSolved Example

Problem: Suppose that there are total 5 baskets of Apples and 6 baskets of Bananas. If Bananas are 15 more than Apples, represent them in algebric form in both the expanded and factorised forms.

Baskets filled with apples and bananas to represent grouping terms in factorisation

Solution: 

If there are x number of apples in one basket then bananas will be x + 15, And we can represent the total number of Apples and Bananas in all the baskets as follows:

Step-by-step example of expanding 5x + 6(x + 15) to get 11x + 90 and vice versa

certified Physics and Maths tutorSolved Example

Problem: Suppose a person owns x no. of cows and horses 10 more than cows. similarly another person owns the y no. of cows and horses which are 15 more than the cows he owns. Then represent the total number of cows and horses these two persons own.

Cartoon of cows, horses, and farmers illustrating a concept from expanding and factorising

Solution: 

It can be represented as:

Expansion and factorisation example showing expressions with x and y variables

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