Complete the Square (Part 1 of Method)

Question

What is the intermediate step in the form (x+a)^2=b as a result of completing the square for the following equation?

-3x^2=18x+102

Answer 1

0 = 3x² + 18x + 102

0 = x² + 6x + 34

0 = x² + 6x + 9 + 34 - 9

Answer 2

First, let's move everything to the left side of the equals:

-3x^2 - 18x - 102 = 0

For the sake of simplicity, we can divide everything by -3 and obtain:

x^2 + 6x + 34 = 0

To complete the square, we must add (and also subtract!) a (b/2)^2. b = 6 in this case, so we have:

x^2 + 6x + (6/2)^2 - (6/2)^2 + 34 = 0.

Simplifying:

x^2 + 6x + 3^2 - 3^2 + 34 = 0

x^2 + 6x + 9 + 25 = 0

The point of using this method is that we can now factor (x^2 + 6x + 9) into (x+3)^2. So, our equation turns into:

(x+3)^2 + 25 = 0

Or:

(x+3)^2 = -25, which you'll find to be the form you are looking for.

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