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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in October 2020. This work is licensed under a Creative Commons BY 4.0 International License.

Solving Quadratic Equations by Completing the Square


Completing the square is another method for solving quadratic equations. We consider a general quadratic equation in which the coefficient of the leading term is not equal to one, in other words: 


When we try to solve a quadratic equation by completing the square, we are looking for an equation of the form:


where h and d are constants. The completing the square method involves the following steps:


Step 1) Divide all terms by the coefficient of .
Step 2) Find 
Step 3) Find 
Step 4) Add  to both sides of the equation.
Step 5) Complete the square on the left-hand-side of the equation.

Step 6) Keep the squared term on the left-hand-side and the constants on the right-hand-side.

Step 7) Take the square root of both sides and solve for the variable.


Example: Solve by completing the square.


Step 1) Divide all terms by 3

Step 2) Divide the coefficient of x by 2: 

Step 3) Square the number from Step 2):

Step 4) Add the number from Step 3) to both sides of the equation:

Step 5) Complete the square on the left-hand-side:
Step 6) Isolate the squared term on the left-hand-side of the equation:

Step 7) Take the square root of both sides and solve:




Check out this presentation by our Math Tutor Dr. Shohreh Rahmati about solving quadratic equations by completing the square.

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