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

1 Answer
Jun 22, 2015

Answer:

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)#