smomofo has it right - this is a method called "completing the square." essentially the goal is to add the same constant to both sides so as to turn the quadratic in x into a perfect square.

ex: [some function of y] = x^2 + 4x

add 4 to both sides: [some function of y] + 4 = x^2 +4x +4 = (x+2)^2

now you can sqrt both sides: sqrt( [function of y] +4) = x+2

and subtract 2, and rearrange: x = sqrt([function of y]+4) - 2

in general, if the square you are completing is x^2 + cx then you will want to add (c/2)^2 to both sides, and the quadratic can be rewritten (x+c/2)^2.

note: i just looked back at your algebra and noticed that your completing of the square looked ok - you seem to understand the idea.

however, when you had y^2 = -(x^2-2x) you added 1 to the left, and added 1 to the right INSIDE the parentheses, which has a -1 coefficient. essentially, this is like adding one to the left side, and subtracting one from the right side. be careful!

first divide both sides by -1 so the x^2 term has +1 as its coefficient, THEN add 1 to both sides. this will give you the proper 1-y^2 on the left, or possibly written, -y^2 + 1 (which is equally correct)