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)