Completing the Square

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Completing the square of a quadratic function is useful for solving the equation , for plotting the graph of the function, or for finding the center of the circle (or other conic section) that is the graph of .

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Here is how completing the square works in words:

(1) Factor out from the first two terms, .

(2) Add the constant term in order to be able to factor as a square. Subtract from to keep equality.

(3) Factor to get the result, .

Here it is in symbols:

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Contributed by: George Beck (March 2011)
Open content licensed under CC BY-NC-SA


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In calculus, completing the square lets you use substitution to change integrals involving quadratics to some standard form. For example,

and or .



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