Methods of Factorisation (SQA National 5 Maths): Revision Note

Factorising using a combination of methods

How do I know if an expression factorises?

• The easiest way to check if ax² + bx + c factorises is to check if you can find a pair of integers which:

  • Multiply to give ac

  • Sum to give b

  • If you can find integers to satisfy this, the expression must factorise

• There are some alternate methods to check:

  • Method 1: Use a calculator to solve the quadratic expression equal to 0

    • Only some calculators have this functionality

    • If the solutions are integers or fractions (without square roots), then the quadratic expression will factorise

  • Method 2: Find the value under the square root in the quadratic formula

    • b² – 4ac

      • This is known as the discriminant

    • If this number is a square number, then the quadratic expression will factorise

Which factorisation method should I use for a quadratic expression?

• Does it have 2 terms only?

  • Yes, like 

    • Factorise out the highest common factor, x

    • 

  • Yes, like 

    • Use the difference of two squares to factorise

    • 

• Does it have 3 terms?

  • Yes, starting with x² like

    • Use the methods in factorising quadratics x²+bx+c, for example by finding two numbers that add to -3 and multiply to -10

    • 

  • Yes, starting with ax² like

    • Check to see if the 3 in front of x² is a common factor for all three terms (which it is in this case), then factorise it out of all three terms

    • 

    • The quadratic expression inside the brackets is now x² +... , which factorises more easily

    • 

  • Yes, starting with ax² like

    • The 3 in front of x² is not a common factor for all three term

    • Use the methods in factorising quadratics ax²+bx+c, for example factorising by grouping or factorising using a grid

    • 

  • It can also be useful to spot a quadratic in the form

    • This factorises to

    • E.g.