Deciding the Quadratic Factorisation Method (Edexcel IGCSE Maths B): Revision Note

Quadratics factorising 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

    • 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 $x^2-7x$

    • Factorise out the highest common factor, x

    • $x\left(x-7\right)$

  • Yes, like $x^2-9$

    • Use the "difference of two squares" to factorise

    • $\left(x+3\right)\left(x-3\right)$

Does it have 3 terms?

• Yes, starting with x² like $x^2-3x-10$

  • Use "factorising simple quadratics" by finding two numbers that add to -3 and multiply to -10

  • $\left(x+2\right)\left(x-5\right)$

• Yes, starting with ax² like $3x^2+15x+18$

  • 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

  • $3\left(x^2+5x+6\right)$

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

  • $3\left(x+2\right)\left(x+3\right)$

• Yes, starting with ax² like $3x^2-5x-2$

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

  • Use "factorising harder quadratics", for example factorising by grouping or factorising using a grid

  • $\left(3x+1\right)\left(x-2\right)$

What other expressions should I be able to factorise?

• You may have a cubed term like $x^3-3x^2-10x$

  • Check to see if x is a common factor for all three terms (which it is in this case), so factorise it out of all three terms

  • $x\left(x^2-3x-10\right)$

  • The remaining quadratic can then be factorised

  • $x\left(x+2\right)\left(x-5\right)$

• It can also be useful to spot a quadratic in the form $x^2+2ax+a^2$

  • This factorises to ${\left(x+a\right)}^2$

  • E.g. $x^2+6x+9={\left(x+3\right)}^2$