Question

How do you solve `x^2 + 6x +8= 0` by completing the square?

Answer 1

`x=-4 or x=-2`

(see below for method)

Explanation:

Given:
`color(white)("XXX")x^2+6x+8=0`

It will be convenient (although not necessary) to move the constant to the right side as:
`color(white)("XXX")color(blue)(x^2+6x=-8)`

Completing the square

For the general squared binomial: `(x+a)^2 = x^2+2a+a^2`

So if `x^2+6x`
`color(white)("XXX")`are the first two terms of an expanded squared binomial
then `a=3 and a^2=9`
and we need to add `9` (to both sides) to complete the square

`color(blue)(x^2+6x)color(red)(+9) = color(blue)(-8)color(red)(+9)`

`color(green)((x+3)^2= 1)`

If `(x+3)^2=1`
then
`color(white)("XXX")x+3=+-1`

`color(white)("XXX")x=-3+-1`

`color(white)("XXX")x=-4 or x=-2`