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Completing the Square (Edexcel IGCSE Further Pure Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Completing the Square

How can I rewrite the first two terms of a quadratic expression as the difference of two squares?

  • Look at the quadratic expression x squared plus b x plus c 

  • The first two terms can be written as the difference of two squares using the following rule

x squared plus b x is the same as open parentheses x plus p close parentheses squared minus p squared where p is half of b

  • Check this is true by expanding the right-hand side

    • Is x squared plus 2 x the same as open parentheses x plus 1 close parentheses squared minus 1 squared?

      • Yes: open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses minus 1 squared equals x squared plus 2 x plus 1 minus 1 equals x squared plus 2 x

  • This works for negative values of bold italic b too

    •  x squared minus 20 x can be written as open parentheses x minus 10 close parentheses squared minus open parentheses negative 10 close parentheses squared which is open parentheses x minus 10 close parentheses squared minus 100

    • A negative b does not change the sign at the end

         

How do I complete the square?

  • Completing the square is a way to rewrite a quadratic expression in a form containing a squared bracket

  • To complete the square on x squared plus 10 x plus 9

    • Use the rule above to replace the first two terms, x squared plus 10 x, with open parentheses x plus 5 close parentheses squared minus 5 squared

    • add 9:  open parentheses x plus 5 close parentheses squared minus 5 squared plus 9

    • simplify the numbers:  open parentheses x plus 5 close parentheses squared minus 25 plus 9

    • answer: open parentheses x plus 5 close parentheses squared minus 16

How do I complete the square when there is a coefficient in front of the x2 term?

  • You first need to take bold italic a out as a factor of the x squared and x terms only

    • a x squared plus b x plus c equals a open square brackets x squared plus b over a x close square brackets plus c

      • Use square-shaped brackets here to avoid confusion with round brackets later

    • For example,  4 x squared plus 16 x plus 5 space equals space 4 open square brackets x squared plus 4 x close square brackets plus 5

  • Then complete the square on the bit inside the square brackets: x squared plus b over a x

    • This gives a open square brackets open parentheses x plus p close parentheses squared minus p squared close square brackets plus c

      • where p is half of b over a

    • 4 open square brackets x squared plus 4 x close square brackets plus 5 space equals space 4 open square brackets open parentheses x plus 2 close parentheses squared minus 4 close square brackets plus 5

  • Finally multiply this expression by the a outside the square brackets and add the c

    • a open parentheses x plus p close parentheses squared minus a p squared plus c

    • This looks far more complicated than it is in practice!

      • Usually you are asked to give your final answer in the form  a open parentheses x plus p close parentheses squared plus q 

      • Here  q equals negative a p squared plus c

    • 4 open square brackets open parentheses x plus 2 close parentheses squared minus 4 close square brackets plus 5 space equals space 4 open parentheses x plus 2 close parentheses squared minus 16 plus 5 space equals space 4 open parentheses x plus 2 close parentheses squared minus 11

  • For quadratics like bold minus bold italic x to the power of bold 2 bold plus bold italic b bold italic x bold plus bold italic c, do the above with a equals negative 1

     

How do I find the turning point by completing the square?

  • Completing the square helps us find the turning point on a quadratic graph

    • If y equals open parentheses x plus p close parentheses squared plus q then the turning point is at open parentheses negative p comma q close parentheses

      • Notice the negative sign in the x-coordinate

      • This links to transformations of graphs (translating y equals x squared by p to the left and q up)

    • If y equals a open parentheses x plus p close parentheses squared plus q then the turning point is still at open parentheses negative p comma q close parentheses

      • It's a minimum point if  a greater than 0

      • It's a maximum point if  a less than 0

  • It can also help you create the equation of a quadratic when given the turning point

Completing the square Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

  • It can also be used to prove and/or show results using the fact that any "squared term", i.e. the bracket (x ± p)2 , will always be greater than or equal to 0

    • You cannot square a number and get a negative value

Completing the square Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

Examiner Tips and Tricks

  • Expand your answer to check that you have completed the square correctly.

Worked Example

(a) By completing the square, find the coordinates of the turning point on the graph of y equals x squared plus 6 x minus 11.

Find half of plus 6 (call this p)
 

p equals 6 over 2 equals 3
 

Write x squared plus 6 x in the form open parentheses x plus p close parentheses squared minus p squared 
 

x squared plus 6 x is the same as open parentheses x plus 3 close parentheses squared minus 3 squared
 

Put this result into the equation of the curve
 

y equals open parentheses x plus 3 close parentheses squared minus 3 squared minus 11
 

Simplify the numbers
 

y equals open parentheses x plus 3 close parentheses squared minus 20
 

Use the fact that the turning point of y equals open parentheses x plus p close parentheses squared plus q is at open parentheses negative p comma q close parentheses

Here p equals 3 and q equals negative 20

Turning point at begin bold style stretchy left parenthesis negative 3 comma space 20 stretchy right parenthesis end style

(b) Write negative 3 x squared plus 12 x plus 24 in the form a open parentheses x plus p close parentheses squared plus q
 

Factorise negative 3 out of the first two terms only
Use square-shaped brackets
 

negative 3 open square brackets x squared minus 4 x close square brackets plus 24
 

Complete the square on the x squared minus 4 x inside the brackets (write in the form open parentheses x plus p close parentheses squared minus p squared where p is half of negative 4)
 

negative 3 open square brackets open parentheses x minus 2 close parentheses squared minus open parentheses negative 2 close parentheses squared close square brackets plus 24
 

Simplify the numbers inside the brackets
open parentheses negative 2 close parentheses squared is 4
 

negative 3 open square brackets open parentheses x minus 2 close parentheses squared minus 4 close square brackets plus 24
 

Multiply all the terms inside the square-shaped brackets by negative 3
 

negative 3 open parentheses x minus 2 close parentheses squared plus 12 plus 24
 

Simplify the numbers
 

negative 3 open parentheses x minus 2 close parentheses squared plus 36
 

This is now in the form a open parentheses x plus p close parentheses squared plus q where a equals negative 3, p equals negative 2 and q equals 36

bold minus bold 3 stretchy left parenthesis x minus 2 stretchy right parenthesis to the power of bold 2 bold plus bold 36

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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.