 # 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 x 2 − 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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