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Completing the Square

This is a method by which a quadratic expression is rewritten as the difference of the square of a binomial and a constant. To complete the square, there are five steps to follow.

1

Factor Out the Coefficient of

It is easier to complete the square when the expression is written in the form Therefore, if the coefficient of is not it should be factored out. For the simplicity of the following steps, it will be assumed that For values of other than the same steps should be performed. The only difference is that the new coefficients — instead of and instead of — will be used.

2

Identify the Constant Needed to Complete the Square

The constant needed to complete the square can now be identified by focusing on the term, while ignoring the rest. One way to find this constant is by squaring half the coefficient of the term, which in this case is Note that leaving the constant as a power makes the next steps easier to perform.

3

Complete the Square

The square can now be completed by adding and subtracting the constant found in Step Note that the value of the original expression will not be changed since the result of adding and subtracting the same value is equal to The first three terms form a perfect square trinomial, which can be factored as the square of a binomial. The other two terms do not contain the variable so their value is constant.

4

Factor the Perfect Square Trinomial
The perfect square trinomial can now be factored and rewritten as the square of a binomial.
The process of completing the square is now finished.

5

Simplify the expression
Finally, if needed, the expression can be simplified. In case was not equal to now is a good time to remove the parentheses and multiply the obtained expression by
Derive function
Simplify
${\textstyle a\left(x+\dfrac{b}{2a}\right)^2-\cancel{{\color{#0000FF}{a}}}\cdot \dfrac{b^2}{4a^\cancel{{\color{#0000FF}{2}}}}+\cancel{{\color{#0000FF}{a}}}\cdot \dfrac{c}{\cancel{{\color{#0000FF}{a}}}}}$

Extra

The method of completing the square is often used to solve quadratic equations. To do so, the resulting square of a binomial should be written on one side of the equation, while the constant should be on the opposite side. Then, the solutions are found by taking the square root of each side of the equation.

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Why

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Instead of a purely algebraic approach, it can be helpful to visualize the process of completing the square geometrically.
The animation shows that the process of completing the square can be understood as finding the dimensions of the that completes an incomplete square.
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Further explanation about a difficult or interesting topic.

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