Completing the Square (DP IB Analysis & Approaches (AA)): Revision Note

Completing the square

Why is completing the square for quadratics useful?

• Completing the square gives the maximum or minimum of a quadratic function

  • This can be used to define the range of the function

• It gives the vertex when drawing the graph

• It can be used to solve quadratic equations

• It can be used to derive the quadratic formula

How do I complete the square for a monic quadratic of the form x²+bx+c?

• Halve the value of and write down

  • This is because 

    • so it is similar to

• Subtract the unwanted  term and add on the constant

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• For example,

  • 

  • 

How do I complete the square for a non-monic quadratic of the form ax²+bx+c?

• Factorise out the from the terms involving

  •  

  • Leaving the un-factorised will avoid working with lots of fractions

• Complete the square on the quadratic term inside the bracket

  • Halve and write 

    • This is because 

  • Subtract the unwanted  term

• Multiply through by and add on

  • 

  • 

• For example,

  • 

  • 

  • 

How do I use completing the square to optimise with quadratics?

• You can use the completed square form to find the maximum or minimum of an expression

  • The optimal value of is

    • It is a maximum if is positive

    • It is a minimum if is negative

  • The optimal value occurs when

• For example, suppose is the profit in dollars when units are sold

  • The maximum profit is $3000

  • This is achieved when 50 units are sold