Factorising Quadratics
How can I factorise simple quadratics?
- If there is no constant term then just factorise out (a multiple of)
- Factorise quadratics of the form by inspection
- Find a pair of numbers, and , that multiply to give and add to give
- E.g. for the numbers would be and
- The quadratic will factorise as
- So
- Find a pair of numbers, and , that multiply to give and add to give
How can I factorise harder quadratics?
- A harder quadratic is of the form where is not equal to 1 (or 0)
- E.g.
- These can also be factorised by inspection
- This requires a lot of practice and there are no simple rules to follow
- They can be factorised reliably by grouping
- Find a pair of numbers that multiply to and add to
- For , and
- So the two numbers are and
- Rewrite the middle term using those two numbers
- Group and factorise the first two terms and the last two terms by pulling out common factors
- Those two terms now have a common factor (in brackets) that can be factorised out
- Find a pair of numbers that multiply to and add to
How do I factorise a difference of two squares
- A difference of two squares refers to any expression of the form
- I.e. 'something squared subtracted from something else squared'
- For example,
- which is equal to
- Such expressions will factorise as
- This is because
- So
- which is equal to
- which is equal to
Exam Tip
- As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.
- You should be able to recognise a difference of squares in both factorised and unfactorised form
Worked example
We will factorise by inspection
We need two numbers that multiply to and add to
and satisfy this
Write down the brackets
We will factorise by splitting the middle term and grouping
We need two numbers that multiply to and add to
and satisfy this
Split the middle term
Factorise out of the first two terms, and out of the last two terms
These have a common factor of which can be factored out
Recognise that this is a difference of two squares, because
Use the relation