Factorising (SQA National 5 Maths): Revision Note

Solving a quadratic equation by factorising

How do I solve a quadratic equation using factorisation?

• If necessary, rearrange it into the form ax² + bx + c = 0

  • Zero must be on one side

  • It is easier if you rearrange so that a is positive

• Factorise the quadratic and solve each bracket equal to zero

  • If (x + 4)(x - 1) = 0, then either x + 4 = 0 or x - 1 = 0

    • Because if two things multiply together to give zero,

      • then one or the other of them must be equal to zero

• To solve

  • …solve first bracket = 0:

    • x – 3 = 0 

    • add 3 to both sides: x = 3

  • …and solve second bracket = 0

    • x + 7 = 0

    • subtract 7 from both sides: x = -7

  • The two solutions are x = 3 or x = -7
    

    • The solutions in this example are the numbers in the brackets, but with opposite signs

What if there are numbers in front of the x's in the brackets?

• The process is the same

  • There's a bit more work to find the solutions

  • You can't just write down the answers by changing the signs

• To solve 

  • …solve first bracket = 0

    • 2x – 3 = 0

    • add 3 to both sides: 2x = 3

    • divide both sides by 2: x = 

  • …solve second bracket = 0

    • 3x + 5 = 0

    • subtract 5 from both sides: 3x = -5

    • divide both sides by 3: x = 

  • The two solutions are x =  or x = 

What if x is a factor?

• The process is the same

  • Just be sure to handle the x correctly

  • That 'x as a factor' gives one of the solutions

• To solve

  • it may help to think of x as (x – 0) or (x)

  • …solve first bracket = 0 

    • (x) = 0, so x = 0

  • …solve second bracket = 0

    • x – 4 = 0

    • add 4 to both sides: x = 4

  • The two solutions are x = 0 or x = 4

How can I use my calculator to help with solving quadratics by factorising?

• On a calculator paper, you can use your calculator to help you to factorise

  • E.g. a calculator gives solutions to as x =   and x = 

    • Reverse the method above to factorise!

    • 

  • Be careful: a calculator also gives solutions to 12x² + 2x – 4 = 0 as x =  and x = 

    • But 12x² + 2x – 4 ≠ 

    • The right-hand side expands to 6x² + ... not 12x² + ...

    • Multiply outside the brackets by 2 to correct this

    • 12x² + 2x – 4 =

• However quadratic equations on a calculator paper often cannot be solved by factorisation

  • You will need to use the quadratic formula instead