Originally Posted by
SlipEternal What do you mean by "cancelled"? The solution provided by romsek in the original post does not involve any "cancelling". He squared both sides. The problem with squaring both sides (when both sides are real) is that you introduce the possibility that one of the sides was negative while the other positive. This is similar to the issue with complex numbers (where squaring may provide more solutions than actually exist). Squaring both sides over the complex numbers is still possible, and any result from that must still hold for the "unsquared" solution. So, squaring both sides gives a necessary condition for a solution, but it does not give a sufficient solution. So, let's look for additional solutions over the complex numbers. We know that either $a$ or $b$ must be zero (that is still a necessary condition for a solution). Now, we want to check if $b=0$, what possible complex values are available for $a$:
$$\sqrt{(x+iy)^2}=x+iy$$
This is true when $x\ge 0$.