Question

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

Answer 1

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)`