Solve for $x$: $\frac{\sqrt{x}}{2} = -1$

Question

I recently saw in the Thread on Mathematical Misconceptions a post which implied that the following equation has no known solutions: $$\frac{\sqrt{x}}{2} = -1$$ Why does this not have any solutions? It seems to me that $\sqrt{x} = -2$, so $x = (-2)(-2) = 4$. Does it not have any solutions because $\sqrt{x}$ is explicitly defined as $-2$, and if $x$ were $4$, then $\sqrt{x}$ would have to equal $\pm 2$?

Answer 1

The symbol $\sqrt{x}$ refers to the unique non-negative number whose square is $x$. There is no ambiguity in terms of signs for the symbol $\sqrt{x}$.