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Quadratic Equation Methods (Cambridge O Level Maths)

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Mark

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Mark

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Quadratic Equation Methods

If you have to solve a quadratic equation but are not told which method to use, here is a guide as to what to do

  

When should I solve by factorisation?

  • When the question asks to solve by factorisation
    • For example, part (a) Factorise 6x2 + 7x – 3, part (b) Solve  6x2 + 7x – 3 = 0
  • When solving two-term quadratic equations
    • For example, solve x2 – 4x = 0
      • …by taking out a common factor of x to get x(x – 4) = 0
      • ...giving x = 0 and x = 4
    • For example, solve x2 – 9 = 0
      • …using the difference of two squares to factorise it as (x + 3)(x – 3) = 0
      • ...giving x = -3 and x = 3
      • (Or by rearranging to x2 = 9 and using ±√ to get x =  = ±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 36x2 + 33x – 20 = 0 using the quadratic formula then by factorisation
  • If in doubt, use the quadratic formula - it always works

   

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 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

Examiner Tip

  • Calculators can solve quadratic equations so use them 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, 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 = 6.70156… or 0.2984...
 

Round your final answers to 2 decimal places

x = 6.70 or x = 0.30

(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, 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

x = bold 9 over bold 2  or x = 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

x = bold 9 over bold 2  or x = 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 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

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 plus-or-minus 2 minus 3
 

Find each solution separately using + first, then - second

x = - 5, x = - 1

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

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    Mark

    Author: Mark

    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.