GCSEMathsAQAHigherRevision Notes2. AlgebraQuadratic EquationsDeciding the Quadratic Method

Deciding the Quadratic Method (AQA GCSE Maths): Revision Note

Exam code: 8300

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

AQA Maths: HigherWJEC Eduqas Maths: HigherOCR Maths: HigherEdexcel Maths: Higher

Deciding the quadratic method

When should I solve by factorisation?

  • Use factorisation when the question asks to solve by factorisation

    • For example

      • part (a) Factorise 6x2 + 7x – 3

      • part (b) Solve  6x2 + 7x – 3 = 0

  • Use factorisation when solving two-term quadratic equations

    • For example, solve x2 – 4x = 0

      • Take out a common factor of x to get x(x – 4) = 0

      • So x = 0 and x = 4

    • For example, solve x2 – 9 = 0

      • Use the difference of two squares to factorise it as (x + 3)(x – 3) = 0

      • So x = -3 and x = 3

      • (Or rearrange to x2 = 9 and use ±√ to get x = ±3)

  • Factorising can often be the quickest way to solve a quadratic equation

When should I use the quadratic formula?

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

    • This is a hint that the equation will not factorise

  • Use the quadratic formula when it may be faster than factorising

    • It's quicker to solve 36x2 + 33x – 20 = 0 using the quadratic formula than by factorisation

  • Use the quadratic formula if in doubt, as it always works

When should I solve by completing the square?

  • Use 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

  • Use completing the square when making x the subject of harder formulae containing both x2 and x terms

    • For example, make x the subject of the formula x2 + 6x = y

      • Complete the square: (x + 3)2 – 9 = y

      • Add 9 to both sides: (x + 3)2 = y + 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

  • Completing the square always works

    • But it's not always quick or easy to do

Examiner Tips and Tricks

  • If your calculator solves quadratic equations, use it to check your solutions

  • If the solutions on your calculator are whole numbers or fractions (with no square roots), this means the quadratic equation does factorise

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 = 1, b = -7 and c = 2 into the formula
Put 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 = 6.70156… or 0.2984...

Round your final answers to 2 decimal places

x = 6.70  or x = 0.30 (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 = 16, b = -82 and c = 45 into the formula
Put 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

x = 9 over 2  or x = 5 over 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

x = 9 over 2  or x = 5 over 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 x2 + 6x as (x + p)2 - p2

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 x2 + 6x with (x + 3)2 – 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

Now solve it
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

x = - 1  or  x = - 5

Even though the quadratic factorises to (x + 5)(x + 1), this is not the method asked for in the question

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourselfFlashcards
    Previous:Completing the SquareNext:Linear Simultaneous Equations
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    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

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.