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Solving Quadratic Equations (Edexcel IGCSE Further Pure Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Solving Quadratic Equations

You should be familiar with solving quadratic equations from your IGCSE Mathematics course.
This is a quick revision guide about the different methods and when to use them.  

When should I solve by factorisation?

  • When the question asks to solve by factorisation

    • For example, part (a) Factorise 6 x squared plus 7 x minus 3, part (b) Solve 6 x squared plus 7 x minus 3 equals 0

      • Factorises as  open parentheses 3 x minus 1 close parentheses open parentheses 2 x plus 3 close parentheses equals 0

      • Solutions are  x equals 1 third  and  x equals negative 3 over 2

  • When solving two-term quadratic equations

    • For example, solve x squared minus 4 x equals 0

      • Take out a common factor of x to get x open parentheses x minus 4 close parentheses equals 0

      • Solutions are x equals 0 and x equals 4

    • For example, solve x squared minus 9 equals 0

      • Use difference of two squares to factorise it as open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses equals 0

      • Solutions are x equals negative 3 and x equals 3

      • (Could also rearrange to x squared equals 9 and use ±√ to get x equals plus-or-minus 3)

When should I use the quadratic formula?

  • When the question says to leave solutions correct to a given accuracy (2 decimal places, 3 significant figures etc)

  • When the quadratic formula may be faster than factorising

    • It's quicker to solve 36 x squared plus 33 x minus 20 equals 0 using the quadratic formula than by factorisation

  • If in doubt, use the quadratic formula - it always works

  • You must remember the formula however - it isn't on the exam formula sheet

  • If  a x squared plus b x plus c equals 0,  the solutions are

    • x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

When should I solve by completing the square?

  • When part (a) of a question says to complete the square and part (b) says to use part (a) to solve the equation

  • When making x the subject of harder formulae containing x squared and x terms

    • For example, make x the subject of the formula  x squared plus 6 x equals y

      • Complete the square: open parentheses x plus 3 close parentheses squared minus 9 equals y

      • Add 9 to both sides: open parentheses x plus 3 close parentheses squared equals y plus 9

      • Take square roots and use ±:  x plus 3 equals plus-or-minus square root of y plus 9 end root

      • Subtract 3:  x equals negative 3 plus-or-minus square root of y plus 9 end root

  • Like the quadratic formula, completing the square will always work

    • But it is not always quick or easy to use the method

Examiner Tips and Tricks

  • Some calculators can solve quadratic equations

    • Even if you need to show working you can use a calculator to check your solutions

    • If the calculator solutions are whole numbers or fractions (with no square roots), this means the quadratic can be factorised

Worked Example

(a) Solve x squared minus 7 x plus 2 equals 0, giving your answers correct to 2 decimal places 

“Correct to 2 decimal places” suggests using the quadratic formula
Substitute a equals 1, b equals negative 7 and c equals 2 into the formula, putting brackets around any negative numbers
 

  x equals fraction numerator negative open parentheses negative 7 close parentheses plus-or-minus square root of open parentheses negative 7 close parentheses squared minus 4 cross times 1 cross times 2 end root over denominator 2 cross times 1 end fraction

Use a calculator to find each solution 

x equals 6.7015... space space space space or space space space space x equals 0.2984... 

Round your final answers to 2 decimal places

bold italic x bold equals bold 6 bold. bold 70 bold space bold space bold or bold space bold space bold italic x bold equals bold 0 bold. bold 30  (2 d.p.)

(b) Solve 16 x squared minus 82 x plus 45 equals 0
 

Method 1
If you cannot spot the factorisation, use the quadratic formula
Substitute a equals 16, b equals negative 82 and c equals 45 into the formula, putting brackets around any negative numbers

x equals fraction numerator negative open parentheses negative 82 close parentheses plus-or-minus square root of open parentheses negative 82 close parentheses squared minus 4 cross times 16 cross times 45 end root over denominator 2 cross times 16 end fraction

Use a calculator to find each solution

bold italic x bold equals bold 9 over bold 2  or bold italic x bold equals bold 5 over bold 8

Method 2
If you do spot the factorisation, (2x – 9)(8x – 5), then use that method instead 

open parentheses 2 x minus 9 close parentheses open parentheses 8 x minus 5 close parentheses equals 0 

Set the first bracket equal to zero 

2 x minus 9 equals 0 

Add 9 to both sides then divide by 2 

table row cell 2 x end cell equals 9 row x equals cell 9 over 2 end cell end table

Set the second bracket equal to zero 

8 x minus 5 equals 0 

Add 5 to both sides then divide by 8 

table row cell 8 x end cell equals 5 row x equals cell 5 over 8 end cell end table

bold italic x bold equals bold 9 over bold 2 bold space bold space bold or bold space bold space bold italic x bold equals bold 5 over bold 8

 (c) By writing x squared plus 6 x plus 5 in the form open parentheses x plus p close parentheses squared plus q, solve x squared plus 6 x plus 5 equals 0
 

This question wants you to complete the square first
Find p (by halving the middle number) 

p equals 6 over 2 equals 3 

Write x squared plus 6 x as open parentheses x plus p close parentheses squared minus p squared
 

table row cell x squared plus 6 x end cell equals cell open parentheses x plus 3 close parentheses squared minus 3 squared end cell row blank equals cell open parentheses x plus 3 close parentheses squared minus 9 end cell end table 

Replace x squared plus 6 x with open parentheses x plus 3 close parentheses squared minus 9 in the equation 

table row cell open parentheses x plus 3 close parentheses squared minus 9 plus 5 end cell equals 0 row cell open parentheses x plus 3 close parentheses squared minus 4 end cell equals 0 end table

Make x the subject of the equation
Start by adding 4 to both sides 

open parentheses x plus 3 close parentheses squared equals 4 

Take square roots of both sides (include a ± sign to get both solutions) 

x plus 3 equals plus-or-minus square root of 4 equals plus-or-minus 2 

Subtract 3 from both sides 

x equals negative 3 plus-or-minus 2 

Find each solution separately using + first, then - second

bold italic x bold equals bold minus bold 5 bold space bold space bold or bold space bold space bold italic x bold equals bold minus bold 1

Even though the quadratic factorises to open parentheses x plus 5 close parentheses open parentheses x plus 1 close parentheses, this is not the method asked for in the question

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    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.