Question

How do you solve `x = 3 +sqrt(x^2 - 9)`?

Answer 1

Isolate the radical; square; use normal solution techniques; then check for extraneous results.
Giving: `x=3`

Explanation:

Given: `x =3+sqrt(x^2-9)`

Isolate radical by subtracting 3 from both sides:
`color(white)("XXXX")` `x-3 = sqrt(x^2-9)`
Square both sides
`color(white)("XXXX")` `(x-3)^2 = x-9`
Factor (both sides)
`color(white)("XXXX")` `(x-3)(x-3) = (x+3)(x-3)`
Note potential solution `(x-3 = 0 rarr x=3)`
Divide both sides by `(x-3)` assuming `x!=3`
`color(white)("XXXX")` `color(red)(cancel(color(black)(x)))-3 = color(red)(cancel(color(black)(x)))+3`

`color(white)("XXXX")` `-3 = +3`

So the only possible solution we have found is `x=3`

Checking to ensure that this is not an extraneous root introduced by squaring; replace `x` with `3` in the original equation:
`color(white)("XXXX")` `(3) ?=? 3 +sqrt((3)^2-9)`?
`color(white)("XXXX")`Yes; equality holds
`color(white)("XXXX")` `x=3` is a valid solution