Factorising Quadratics | Edexcel GCSE Maths: Foundation Revision Notes 2017

Factorising Simple Quadratics

What is a quadratic expression?

A quadratic expression is an algebraic expression where the highest power of the unknown is 2
- The general form for a quadratic expression is $a x^{2} + b x + c$
- You will only come across quadratic expressions where $a = 1$
- Remember that $b$ or $c$ could have a value of 0

How do I factorise a quadratic expression?

You can factorise a quadratic through inspection
If c is positive then the factor pair must have either two positive factors or two negative factors
- A positive multiplied by a positive gives a positive
- A negative multiplied by a negative gives a positive
E.g. Factorise
- Identify all the factor pairs of c
  - In this example c = +4
  - Factor pairs for this example include:
    1, 4 or -1, -4
    2, 2 or -2, -2
- Identify the factor pair that add together to give b
  - In this example, b = +5
  - Because of this we know that we are going to use a factor pair where both factors are positive
  - The factor pair that adds to 5 is 1 and 4
- Write a pair of brackets each with x and one of the factors
  - (x + 1)(x + 4)

How do I factorise a quadratic where c is positive but b is negative?

If c is positive then the factor pair must have either two positive factors or two negative factors
- A positive multiplied by a positive gives a positive
- A negative multiplied by a negative gives a positive
E.g. Factorise $x^{2} - 8 x + 15$
- Identify all the factor pairs of c
  - In this example, c = +15
  - Factor pairs of +15 include:
    1, 15 or -1, -15
    3, 5 or -3, -5
- Identify the factor pair that add together to give b
  - In this example, b = -8
  - Because of this we know that we are going to use a factor pair where the factors are both negative
  - The factor pair that adds to -8 is -3 and -5
- Write a pair of brackets each with x and one of the factors
  - (x -3)(x - 5)

How do I factorise a quadratic where c is negative?

If c is negative then the factor pair must have one positive factor and one negative factor
- A positive multiplied by a negative gives a negative
E.g. Factorise $x^{2} - 2 x - 8$
- Identify all the factor pairs of c
  - In this example, c = -8
  - Factor pairs of -8 include:
    -1, +8 or +1, -8
    +2, -4 or -2, +4
- Identify the factor pair that add together to give b
  - In this example, b = -2
  - The factor pair that adds to -2 is +2, -4
- Write a pair of brackets each with x and one of the factors
  - (x + 2)(x - 4)

Examiner Tip

As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.

Worked example

(a) Factorise $x^{2} - 4 x - 21$ .

Factorise the expression by inspection

We need two numbers that multiply to -21 and sum to -4

Identify all factor pairs of -21

+1, - 21
-1, +21
+3, -7
-3, +7

Identify the factor pair that sum to -4

+3, -7

Write down the brackets

(x + 3)(x - 7)

Difference Of Two Squares

What is the difference of two squares?

When a 'squared' quantity is subtracted from another 'squared' quantity, you get the difference of two squares
- for example,
  - a² - b²
  - 9² - 5²

How do I factorise the difference of two squares?

The difference of two squares, a² - b², factorises to
- $(a + b) (a - b)$
You can see why this is if you work backwards and expand the brackets (a + b)(a - b)
- (a + b)(a - b) = a² - ab + ba - b²
  - ab is the same quantity as ba, so -ab and +ba cancel out
- (a + b)(a - b) = a² - b²

Examiner Tip

The word difference in maths means a subtraction, it should remind you that you are subtracting one squared term from another.
The phrase 'difference of two squares' is not often used in exam questions.
- Look out for questions that involve quadratics with only 2 terms and a subtraction sign between them!

Worked example

(a)

Factorise

x^{2} - 16

Recognise that $x^{2}$ and $16$ are both squared terms
Also, the second term is subtracted from the first term

Factorise using the difference of two squares

$x^{2} - 16 = {(x)}^{2} - {(4)}^{2}$

Put the square root of each term added together in the first bracket
Put the square root of each term subtracted from each other in the second bracket

$(x + 4) (x - 4)$

(b)

Factorise

4 x^{2} - 25

Recognise that $4 x^{2}$ and $25$ are both squared terms
Also, the second term is subtracted from the first term

Factorise using the difference of two squares

$4 x^{2} - 25 = {(2 x)}^{2} - {(5)}^{2}$

Put the square root of each term added together in the first bracket
Put the square root of each term subtracted from each other in the second bracket

$(2 x + 5) (2 x - 5)$

Factorising Quadratics (Edexcel GCSE Maths: Foundation)

Revision Note

Factorising Simple Quadratics

What is a quadratic expression?

How do I factorise a quadratic expression?

How do I factorise a quadratic where c is positive but b is negative?

How do I factorise a quadratic where c is negative?

Examiner Tip

Worked example

Difference Of Two Squares

What is the difference of two squares?

How do I factorise the difference of two squares?

Examiner Tip

Worked example

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Author: Mark