the square root of sqrt((x+1)^2+y^2)-sqrt((x-2)^2+y^2) = 1 can be removed by take both side square more than once.
Solve and sketch its graph.
First I notice that I'm finding the complex numbers where the distance between z and -1 = the distance between z and 2 +1 (not sure how the +1 affects things).
here's where I'm not sure where to go because I can't just square both sides to remove the roots with the 1 there. How do I proceed?
If I recall correctly, the figure such that the difference of the distances for each point on the figure to two fixed points is the hyperbola having those fixed points as foci.
I misread your equation. I have edited what I wrote before. The set of points such that the difference of the distances from two fixed point (either d1- d2 or d2- d1) is a constant is indeed a hyperbola. For only d2- d1 a constant, you have one branch of the hyperbola.
There is an implied restriction on x in this working that will cause one of the branches of this hyperbola to be rejected (that branch is an extraneous solution and is introduced by the squaring process). But I don't have the time to read through it. However, it is sufficient to test a simple point on each branch and reject the branch that contains the point that fails to satisfy the original equation.
Personally, I favour the approach of moving a radical to one side first before squaring. The implied restriction is a lot more obvious. If I have time later, I might show the first few steps.