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# Completing the Square

To complete the square for the expression x2 + bx , add , which is the square of
half
the coefficient of x. Consequently,

When solving quadratic equations by completing the square, you must add to both
sides to maintain equality.

Completing the Square: Leading Coefficient is 1

Let’s solve the equation x2 − 6x + 2 = 0 by completing the square.

 x2 + 6x + 2 = 0 Original Equation x2 + 6x = −2 Subtract 2 from both sides Divide the 6 by 2, square it, and then add to both sides x2 + 6x + 9 = 7 Simplify (x + 3)2 = 7 Perfect square trinomial Extract square roots Solutions

Completing the Square: Leading Coefficient is Not 1

Let’s solve the equation 3x2 − 4x − 5 = 0 by completing the square.

If the leading coefficient of a quadratic equation is not 1, you should divide both sides of
the equation by this coefficient before completing the square.

 3x2 − 4x − 5 = 0 Original equation 3x2 − 4x = 5 Add 5 to both sides Divide both sides by 3 Divide by 2, square it, and then add to both sides Perfect square trinomial Extract square roots Solutions

Using a graphing calculator , you can see that the two solutions are approximately
2.11963 and –0.78630, which agree with the two graphical solutions shown below.

Completing the Square: One Term is Not Present

Let’s solve the equation 4x2 − 7x = 0 by completing the square.

As you can see, we have no constant but we will treat the problem the same as if there
was a constant present . We skip the step of moving the constant over to the other side of
the equation and continue on from there.

 4x2 − 7x = 0 Original equation Divide both sides by 4 Divide by 2, square it, and then add to both sides Perfect square trinomial Extract square roots Solutions
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