Manipulating Expressions

Topic Overview

A mathematical expression refers to when numbers, operators and various symbols are grouped together to 'express' or 'show' the value of something. For example:

4 x 9

5x- 6

4y + 2x - 3

are all mathematical expressions.

When manipulating mathematical expressions, you will encounter various mathematical terms:

Equation

An equation is a term used to describe mathematical values which are equal. For example:

x + 3 = 5

This equation is stating that means that whatever is on the left side of the '= ' sign:

x + 3

is the same value of that which is on the right side:

5

Variable, Constant and Coefficient

Within a mathematical equation, there are various different elements to consider.

The term variable is used to describe the letter symbol within an equation, e.g. 'x' or 'y'. These variable symbols are used because we do not yet know their value.

For example, in the equation: 2x + 3 = 5, we do not yet know the value of 'x' until we have solved the equation.

The term constant is used to describe a number on its own. For example, in the equation 2x + 3 = 5, both 3 and 5 are constants because they are not attached to any variables.

The term coefficient is used to describe the amount by which a variable must be multiplied. For example, in the equation 2x + 3 = 5, 2 is the coefficient, because it tells us that the variable 'x' must be multiplied by 2.

Key Concepts

In the new linear GCSE Maths paper, you will be required to manipulate various mathematical expressions. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Manipulate algebraic expressions by collecting like terms and by taking out common factors
  • Multiplying two linear expressions
  • Factorise quadratic expressions including the difference of two squares
  • Simplify rational expressions

Listed below are a series of summaries and worked examples to help you solidify your knowledge about how to solve manipulate various mathematical expressions.

Worked Examples

1-Collecting like terms
During your GCSE maths exam, you may be faced with several long algebraic expressions. In these circumstances, you can simplify these expressions by 'collecting like terms'.

Example
(a) - Simplify the expression: 3x + 7 - 4x + 2 + 8x -6 - x

Solution
(a) - When 'collecting like terms', you must group together all the x values and all the numbers in order to simplify the expression:

3x + 7 - 4x + 2 + 8x -6 - x

All of the x values can be grouped together :

3x - 4x + 8x -x = 11x - 5x = 6x

All of the numbers can be grouped together:

7 + 2 - 6 = 9 - 6 = 3

Therefore, by collecting like terms, you can simplify the expression 3x + 7 - 4x + 2 + 8x -6 - x into: 6x + 3

2- Taking out common factors
By factorising an expression, you can simplify it. This process of 'taking out common factors' is the reverse of expanding brackets.

Example
(a) - Factorise 4y + 6

Solution
(a) - In order to factorise this expression, you must find the highest common factor (HCF) of 4 and 6. One you have done this, you must divide both terms by this number.

4y +6 = [ (4) x (y) ] + [(6)]

= [(2)x (2) x (y)] + [(2) x (3)]

= [(2)x (2y)] + [(2) x (3)]

= 2(2y +3)

The HCF of 4 and 6 is 2, therefore:

4y + 6 can be factorised into : 2(2y + 3)

3-Factorising quadratic expressions
During your GCSE maths exam, you will also be required to factorise quadratic expressions.

Example
(a) - Factorise

\[5{y^2} + 10y\]

Solution
(a) - Much like the previous example, you must find the highest common factor.

\[5{y^2} = 5*y*y\]
\[10y = 2*5*y\]

As a result, the HCF is: 5 x y

Therefore, you must divide both values by 5y:

Therefore

\[5{y^2} + 10y\]

can be factorised into

\[5y(y + 2)\]

Example
(b) - Factorise

\[{x^2} + 6x + 8\]

Solution
In order to factorise this quadratic expression, you must first look for two numbers which add up to make 6 and multiply to make 8:

The factor pairs of 8 are:

1 and 8

2 and 4

2 and 4 add up to make 6.Therefore, you can factorise into:

(x +2) (x+4)

Note: double check your answer by multiplying out the brackets:

\[(x + 2)(x + 4) = {x^2} + 4x + 2x + 8 = {x^2} + 6x + 8\]

Subsequently, you know that your answer is correct!

Example
(c) - Factorise the expression:

\[{y^2} - 5y - 6\]

Solution
(c) - First look for common factor pairs which multiply to make -10 and add up to make -3. The factor pairs of -10 are:

1 and -6

-1 and 6

2 and -3

-2 and 3

Out of these common factor pairs, only 1 and -6 add up to make -5. Therefore, you can factorise into:

\[(y + 1)(y - 6)\]

Once again, double check your answer by multiplying out the brackets:

\[(y + 1)(y - 6) = {y^2} - 6y + y - 6 = {y^2} - 5y - 6\]

4-Factorising the difference of two squares
During your exam, you may also be required to factorise an expression which does not have an 'x' term, for example:

\[{x^2} - 9\]

In these circumstances, you must use the same factorisation method as previously demonstrated, with the exception that you must look for two numbers which will multiply to make 9 and add to make 0.

Example
(a) - Factorise

\[{x^2} - 9\]

Solution
(a) - You need to find two numbers which will multiply to make 9 and add to make 0.

The common factor pairs for 9 are:

1 and -9

-1 and 9

3 and -3

Out of these common factor pairs, only 3 and -3 will add up to make 0. Therefore, this can be factorised into:

\[(x + 3)(x - 3)\]

Note: Not all quadratic expressions without an x term can be factorised. For example:

\[{x^2} - 49\]

will factorise into (x + 7)(x - 7), however

\[{x^2} + 49\]

will not factorise. It is important to remember that, in all examples which will factorise, you must have x^2 minus a square number.

The term which is used to describe the factorisation of these expressions is called 'the difference of two squares'.

Exam Tips

  1. When 'collecting like terms', you must group together all the x values and all the numbers in order to simplify the expression
  2. In order to factorise an expression, you must find the highest common factor (HCF) of the values within the expression. One you have done this, you must divide both terms by this number. This process is the reverse of expanding out the brackets
  3. When factorising quadratic expressions, look for two values which will add up to make the coefficient of the 'x' term and which will multiply to make the value of the constant within the expression.
  4. When factorising the difference of two squares, remember that, in all examples which will factorise, you must have x^2 minus a square number.

Topic Summary

Once you understand the basic principles of manipulating expressions, you will easily be able to factorise and simplify a variety of different expressions. Always remember to double check your answer by multiplying the brackets. By doing so, you can know for sure whether you answer is correct, and if it is not, you can quickly go back and work out the right answer. Above all else, write down all of your working out. This will help you to identify if you have made a mistake and correct it. It will also enable the examiner who is marking your paper to award you the maximum method marks for each question.

Related Topics

  • Solving Equations
  • Change the Subject of a Formula
  • The Straight Line
  • Graphs of Curves
  • Inequalities
  • Sequences
  • Collecting and Using Algebraic Terms
  • Reasoning and Proof
  • Pre-calculus Skills
  • Functions