Double checking problem - simple square root equation?

Question

Problem: √(x+2)-x=0, Note: x+2 is in the square root, -x isn't.
I added x to the right hand column, then squared both √(x-2) and x to get rid of the square root.

This gave me: x+2=x²
I subtracted x and subtracted 2 to get: 0=x²-x-2
I factored to get 0=(x-2)(x+1)
This gave x=2, x=-1

When I redistribute 2 and -1 into the equation -1 doesn't work, but 2 does, so x=2. Is this the final solution?

Answer 1

Your work and answer is correct. I might add that when you get to this point:

sqrt(x+2)=x

You can state the restriction that x >= 0, since x is equal to a square root of some other number. This is what later rules out x=-1 as a solution.

Answer 2

Both 2 and -1 are correct since both will satisfy the equation x^2 - x - 2 = 0

Answer 3

Yes, that's correct. Note that by squaring you allow both the equations
√(x+2) = x
and
√(x+2) = -x
to be satisfied. 2 is the solution to the first equation (the one you want), -1 is the solution to the second equation (which we don't want).

Answer 4

yes it is