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/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?

• Half the value of b and write 

  • This is because 

• Subtract the unwanted  term and add on the constant c

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

• Factorise out the a from the terms involving x

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  • Leaving the c alone will avoid working with lots of fractions

• Complete the square on the quadratic term

  • Half and write 

    • This is because 

  • Subtract the unwanted  term

• Multiply by a and add the constant c

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