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Quadratics Factorising Methods

Quadratics Factorising Methods

How do I know if it factorises?

  • Method 1: Use a calculator to solve the quadratic expression equal to 0
    • If the solutions are integers or fractions (without square roots), then the quadratic expression factorises
  • Method 2: Find the value under the square root in the quadratic formula, b2 – 4ac (called the discriminant)
    • If this number is a perfect square number, then the quadratic expression factorises

 

Which factorisation method should I use for a quadratic expression?

  • Does it have 2 terms only?
    • Yes, like x squared minus 7 x
      • Use "basic factorisation" to take out the highest common factor
      • x open parentheses x minus 7 close parentheses
    • Yes, like x squared minus 9
      • Use the "difference of two squares" to factorise
      • open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses
  • Does it have 3 terms?
    • Yes, starting with x2 like x squared minus 3 x minus 10
      • Use "factorising simple quadratics" by finding two numbers that add to -3 and multiply to -10
      • open parentheses x plus 2 close parentheses open parentheses x minus 5 close parentheses
    • Yes, starting with ax2 like 3 x squared plus 15 x plus 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 use "basic factorisation" to factorise it out first
      • 3 open parentheses x squared plus 5 x plus 6 close parentheses
      • The quadratic expression inside the brackets is now x2 +... , which factorises more easily
      • 3 open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses
    • Yes, starting with ax2 like 3 x squared minus 5 x minus 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
      • open parentheses 3 x plus 1 close parentheses open parentheses x minus 2 close parentheses

Worked example

Factorise  negative 8 x squared plus 100 x minus 48.

 
Spot the common factor of -4 and put outside a set of brackets, work out the terms inside the brackets by dividing the terms in the original expression by -4.

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses

Check the discriminant for the expression inside the brackets, open parentheses b squared minus 4 a c close parentheses, to see if it will factorise.

table row blank blank cell open parentheses negative 25 close parentheses squared minus 4 cross times 2 cross times 12 end cell row blank equals cell 625 minus 96 end cell row blank equals 529 end table

529 equals 13 squared, it is a perfect square so the expression will factorise.

Proceed with factorising 2 x squared minus 25 x plus 12 as you would for a harder quadratic, where a not equal to 1.
"+12" means the signs will be the same.
"-25" means that both signs will be negative.

a cross times c equals 2 cross times 12 equals 24

The only numbers which multiply to give 24 and follow the rules for the signs above are:
open parentheses negative 1 close parentheses cross times open parentheses negative 24 close parentheses and open parentheses negative 2 close parentheses cross times open parentheses negative 12 close parenthesesand open parentheses negative 3 close parentheses cross times open parentheses negative 8 close parentheses and open parentheses negative 4 close parentheses cross times open parentheses negative 6 close parentheses
but only the first pair add to give negative 25.

Split the negative 25 x term into negative 24 x minus x.

table row blank blank cell 2 x squared minus 24 x minus x plus 12 end cell end table

Group and factorise the first two terms, using 2 x as the highest common factor and group and factorise the last two terms using 1 as the highest common factor.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 x open parentheses x minus 12 close parentheses plus 1 open parentheses x minus 12 close parentheses end cell end table

These factorised terms now have a common term of open parentheses x minus 12 close parentheses, so this can now be factorised out.

open parentheses 2 x plus 1 close parentheses open parentheses x minus 12 close parentheses

Put it all together.

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses equals negative 4 open parentheses 2 x minus 1 close parentheses open parentheses x minus 12 close parentheses

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