||z|-2| = z-1
Squaring both sides
|z|^2 - 4|z| + 4 = z^2 - 2z + 1
Since |z|^2 = z^2 we get
-4|z| + 4 = -2z + 1
4|z| = 2z + 3
Again squaring both sides
16z^2 = 4z^2 + 12z + 9
ie 4z^2 - 4z - 3 = 0
Solving for Z we get z = -1/2 or z = 3/2
But when choices are given, checking by substitution will be easier than this long process.
-1/2 is clearly out. When you square an equation, you can get many extraneous roots. Therefore, each root should be plugged to make sure that it does make sense.