Solving Quadratics by Factorising (OCR GCSE Maths) : Revision Note

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 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 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 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 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: x3 over 2

    • …solve second bracket = 0

      • 3x + 5 = 0

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

      • divide both sides by 3: xnegative 5 over 3

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

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 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 (cancel) both sides by x at the beginning

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

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

  • You can use your calculator to help you to factorise

    • A calculator gives solutions to 6 x squared plus x minus 2 equals 0 as xnegative 2 over 3  and x1 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

    • Be careful: a calculator also 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

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

      • Multiply outside the brackets 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

Examiner Tips and Tricks

  • Remember that you can check your solutions by either

    • substituting them back into the original equation

    • using a different quadratic method

    • or using a calculator

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

xnegative 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

x3 over 2

Write both solutions together using “or”

x = negative 7 over 8 or x3 over 2

(c) Solve x open parentheses 5 x minus 1 close parentheses equals 0

Do not divide both sides by(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

x1 fifth

Write both solutions together using “or”

x = 0 or x = 1 fifth

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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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