Question

How do you complete the square to solve `-x² +6x +9=0`?

Answer 1

`-x^2+6x+9=0`
is equivalent to
`color(white)("XXXXX")` `x^2-6x-9=`

to help keep thing simple, move the constant to the right side as
`color(white)("XXXXX")` `x^2-6x = 9`

in the general form of the squared binomial
`color(white)("XXXXX")`
`(x+a)^2 = x^2+2ax+a^2`
so if `x^2-6x` are the first two terms of a squared binomial, `a=3`

and to complete the square, we need to add an extra
`color(white)("XXXXX")` `a^2 = (-3)^2 =9`

Therefore we write
`color(white)("XXXXX")` `x^2-6x+9 = 9 +9`

`color(white)("XXXXX")` `(x-3)^2 = 18`

Taking the square root of both sides:
`color(white)("XXXXX")` `x-3 = +-3sqrt(2)`

and
`color(white)("XXXXX")` `x= 3+3sqrt(2)` or `x=3-3sqrt(2)`