Completing the Square
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An equation in which one side is a perfect square trinomial can be easily solved by taking the square root of each side. 
Consider the example at the right:
                                       

It is this method of solution that is the basis of a process called "completing the square".

Consider the equation:  .
  Our solution strategy will be to "force" a  perfect square trinomial of the form
 
on the left hand side of the equation.
This method of "forcing" the existence of a perfect square trinomial is completing the square.

Steps for Completing the Square:

1.  Be sure that the coefficient of the highest power is one.
If it is not, divide each term by that value to create a leading coefficient of one.

2.  Move the constant term to the right hand side.

3.  Prepare to add the needed value to create the perfect square trinomial.  Be sure to balance the equation.  The boxes may help you remember to balance.
4.  To find the needed value for the perfect square trinomial, take half of the coefficient of the middle term (x-term), square it, and add that value to both sides of the equation.
                      
5.  Factor the perfect square trinomial.
6.  Take the square root of each side and solve.  Remember to consider both plus and minus results.



 

Examples:  Solve each example by completing the square.

1:   

Keep all x related terms on one side.  Move the constant to the right.

Get ready to create a perfect square on the left.  Balance the equation.

Take half of the x-term coefficient and square it.  Add this value to both sides.

Factor and write the perfect square on the left.

Take the square root of both sides.  Be sure to allow for both plus and minus.

Solve for x.

 

 

2:   

This equation needs some re-arranging.

Divide through by 2 to create the leading coefficient of 1 (for p2).

Keep all p related terms on one side.  Move the constant to the right.

Get ready to create a perfect square on the left.  Balance the equation.

Take half of the p-term coefficient and square it.  Add this value to both sides.

Factor and write the perfect square on the left.

Take the square root of both sides.  Be sure to allow for both plus and minus.

Solve for p.  Be sure to express the negative radical as an imaginary number.

 

 

There are other applications of completing the square.  In the section on Equations of Circles, we will see how completing the square is used  to change the equation of a circle from standard form to center-radius form.