How do you solve x^2 + 32x + 15 = -x^2 + 16x + 11 by completing the square?

1 Answer
Alan P.
Jun 22, 2015

Simplify the given equation then apply the process of "completing the squares" to obtain
color(white)("XXXX")x = -4+-sqrt(14)

Explanation:

Given x^2+32x+15 = -x^2+16x+11

First simplify to get all terms involving x on the left side and a simple constant on the right:
color(white)("XXXX")2x^2+16x= -4
Further simplify by dividing by 2
color(white)("XXXX")x^2+8x=-2

Now we are ready to begin completing the square.
#

Noting that the squared binomial (x+a)^2 = x^2+2ax + a^2

If x^2+8x are the first 2 terms of an expanded squared binomial,
then 2ax = 8x rarr a=4 and a^2 =16
So the completed square must be (after remembering that anything added to one side of an equation must also be added to the other)
color(white)("XXXX")x^2+8x+16 = -2+16
or
color(white)("XXXX")(x+4)^2 = 14

Taking the square root of both sides:
color(white)("XXXX")x+4 = +-sqrt(14)
and
color(white)("XXXX")x = -4+-sqrt(14)

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