That is correct you cannot take square roots of negative numbers. Unless you redefine the square root function for a complex co-domain. But since I have never seen it done, and am familar with the construction of numbers. I consider it improper (mathematically illegal) to define it for negative numbers. Go here and try to understand what I say.

Simple.

Theorem: The equation has two solutions when and they are . A unique solution for . And non-real solutions for which are .

Proof: Check that for those solutions satisfy the equation. Furthermore because the complex numbers are a field (a type of algebra) there is at most 2 solutions. Thus those need to be it. If then, since it has no zero divisor (no non-zero numbers that give zero) we conclude that is the only solution.

(NOTE: No negative square roots were used).