rich,

      you have to solve it like this:-

Let z = x+iy.

|z|= x²+y².

|z|-z = (x²+y²) - x -iy and this is = 1+2i.

Equating real and imaginary parts:-

we get, y= -2

and (x²+y²)-x = 1.

put y = -2., take x on the other side and square the whole thing.

we get,

    x²+4 = x²+2x+1

i.e. x=3/2

so, finally z= 3/2 - 2i

 

 

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