Is That a Square Root?

Moderator note:

There are actually 2 solutions to the equation $2^x = x ^ 2$ , namely $x = 2 , 4$ .

Why do we reject the answer of $x = 4$ ?

Hint: Recall that we don't have double implication signs throughout our work. We are assuming that the value converges to $x$ , and then deriving a necessary condition that $x$ must satisfy.
E.g. if we're asked to solve $x = \sqrt{ 2x - 1 }$ , then our chain of implications is
$x = \sqrt{ x + 6 } \Rightarrow x^2 = x + 6 \Leftrightarrow ( x- 3)(x+2) = 0 \Leftrightarrow x = 3, -2$
Because the first implication isn't double sided, we only have a necessary condition, which turns out isn't sufficient. We still have to verify that the final answers satisfy the original equation. In this cause, we reject the answer of $- 2$ since $-2 \neq \sqrt{ -2 + 6 }$ .
Similarly, for the original problem, how do we "verify that the final answers satisfy the original equation"? Clearly at most 1 answer is correct, so which one is it?