Well that's one way to show it, but I'd compare it to killing a fly with a nuclear bomb!
Every complex number can be written as (polar form). Then and are the two square roots of ( denotes the usual, positive square root of the real number ). I'll let you show that they are distinct when . That there cannot be any others is a consequence of the fact that polynomials over a field (such as ) cannot have more roots than their degree.
Yet another way. By DeMoivres' theorem, the kth roots of the complex number are of the form for integer j.
Here, k= 2 so the square roots are given by .
If j is any even number, say j= 2n, so that
If j is odd, say j= 2n+ 1, so .
The only two square roots are and .