Question

How do you solve using the completing the square method `x^2+8x=9`?

Answer 1

`x=1` or `x=9`
(see below for method of completing the square).

Explanation:

A general squared binomial has the relation:
`color(white)("XXX")(a+b)^2=a^2+2ab+b^2`

If `x^2+8x` are the first two terms of an expanded squared binomial
then `a=1` and `b=4`
and the third term needed to complete the square would be `b^2=16`.

If we add `16` to the left side to complete the square there
we will need to add `16` to the right side to maintain the equality.
`color(white)("XXX")x^2+8x+16=9+16`

Re-writing the left side as a squared binomial and simplifying the right side:
`color(white)("XXX")(x+4)^2=25`

If we take the square root of both sides:
`color(white)("XXX")(x+4)=+-5`

which implies:
either `x=1` or `x=-9`