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Solving Quadratics by Factorising (Cambridge (CIE) O Level Additional Maths): Revision Note

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Solving Quadratics by Factorising

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 to use the side where 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 A × B = 0, then either A = 0 or B = 0

  • To solve open parentheses x minus 3 close parentheses open parentheses x plus 7 close parentheses equals 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 have the opposite signs to the numbers in the brackets

  • To solve open parentheses 2 x minus 3 close parentheses open parentheses 3 x plus 5 close parentheses equals 0

    • …solve “first bracket = 0”

      • 2x – 3 = 0

      • add 3 to both sides: 2x = 3

      • divide both sides by 2: x = 3 over 2

    • …solve “second bracket = 0”

      • 3x + 5 = 0

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

      • divide both sides by 3: x = negative 5 over 3

    • The two solutions are x = 3 over 2 or x = negative 5 over 3

  • To solve x open parentheses x minus 4 close parentheses equals 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 both sides by x at the beginning - you will lose a solution (the x = 0 solution)

Examiner Tips and Tricks

  • Where permitted, and if you calculator has a quadratic solving feature, you can use it to check your final solutions!

    • Such calculators also help you to factorise (if you're struggling with that step)

    • e.g.  A calculator gives solutions to 6 x squared plus x minus 2 equals 0 as x = negative 2 over 3  and x = 1 half

      • "Reverse" the method above to factorise!

      • 6 x squared plus x minus 2 equals open parentheses 3 x space plus space 2 close parentheses open parentheses 2 x space minus space 1 close parentheses

    • Warning: a calculator (correctly) gives solutions to 12x2 + 2x – 4 = 0 as x = negative 2 over 3 and x = 1 half

      • But 12x2 + 2x – 4 ≠ open parentheses 3 x plus 2 close parentheses open parentheses 2 x minus 1 close parentheses as these brackets expand to 6x2 + ... not 12x2 + ...

      • Multiply by 2 to correct this

      • 12x2 + 2x – 4 = 2 open parentheses 3 x plus 2 close parentheses open parentheses 2 x minus 1 close parentheses

Worked Example

(a) Solve open parentheses x minus 2 close parentheses open parentheses x plus 5 close parentheses equals 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”

x = 2 or x = -5

  

(b) Solve open parentheses 8 x plus 7 close parentheses open parentheses 2 x minus 3 close parentheses equals 0
 

Set the first bracket equal to zero

8x + 7 = 0

Subtract 7 from both sides

8x = -7

Divide both sides by 8

x = negative 7 over 8

Set the second bracket equal to zero

2x - 3 = 0

Add 3 to both sides

2x = 3

Divide both sides by 2

x = 3 over 2

 

Write both solutions together using “or”

x = bold minus bold 7 over bold 8 or x = bold 3 over bold 2

(c) Solve x open parentheses 5 x minus 1 close parentheses equals 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

5x - 1 = 0

Add 1 to both sides

5x = 1

Divide both sides by 5

x = 1 fifth

Write both solutions together using “or”

x = 0 or x = bold 1 over bold 5

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