ASMaths: PureAQARevision Notes2. Algebra & Functions2.2 Quadratics2.2.3 Completing the square

Completing the square (AQA AS Maths: Pure)

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Completing the square

What is completing the square?

  • Completing the square is another method used to solve quadratic equations
  • It simply means writing y space equals a x squared plus b x plus c in the form y space equals space a left parenthesis x plus p right parenthesis squared plus q
  • It can be used to help find other information about the quadratic like coordinates of the turning point 

How do I complete the square?

The method used will depend on the value of the coefficient of the x2 term in y space equals a x squared plus b x plus c

When a = 1
  • p is half of the coefficient of b
  • q is c - p2

Completing the square Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

 

When a ≠ 1 

  • You first need to take a out as a factor of the x2 and x terms
  • Then continue as above

Completing the square Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

 

When is completing the square useful?

  • Completing the square helps us find the turning point on a quadratic graph
  • 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 a squared term will always be greater than or equal to 0

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

Examiner Tip

  • Sometimes the question will explicitly ask you to complete the square
  • Sometimes it will even remind you of the form to write it in
  • But sometimes it will expect you to spot that completing the square is what you need to do to help with other parts of the question... like finding turning points!

Worked example

Completing the square Example Diagram 1, A Level & AS Level Pure Maths Revision NotesCompleting the square Example Diagram 2

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.