Solving Quadratic Equations (Edexcel IGCSE Maths)

Solving Quadratics by Factorising

Spec point — 2.7A

How do I solve a quadratic equation using factorisation?

• 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 (x−3)(x+7)=0

  • …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 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 x (x−4)=0

  • 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

• It is a common mistake to divide (cancel) both sides by x at the beginning

  • If you do this you will lose a solution (the x = 0 solution)

Exam Hint Remember that you can check your solutions by substituting them back into the original equation.

Worked Example (a) Solve $x^{2}+3x-10=0$ by factorising.

Answer:

Factorise the left-hand side

(x – 2)(x + 5)=0*

Set the first bracket equal to zero

x – 2 = 0*

Add 2 to both sides

x = 2*

Set the second bracket equal to zero

x + 5 = 0*

Subtract 5 from both sides

x = -5*

Write both solutions together using “or”

Final answer: x = 2 or x = -5

(b) Solve $x^{2}-x=0$

Answer:

Factorise the left-hand side

x(x - 1)=0*

*Do not divide both sides by x *(this will lose a solution at the end) Set the first “bracket” equal to zero

(x) = 0

*Solve this equation to find *x

x = 0*

Set the second bracket equal to zero

x - 1 = 0*

Add 1 to both sides

x = 1*

Write both solutions together using “or”

Final answer: x = 0 or x =** 1**