We know that the square root function ($\sqrt{}$) always produces non-negative numbers. Why then, does the square root function produce a negative value here?
$$x^2=4$$
$$\implies \pm \sqrt{x^2}=2$$
$\implies \sqrt{x^2}=2...(i)$, or, $\implies \sqrt{x^2}=-2...(ii)$
In equation (ii), the square root function produces a negative value. I can't even graph equation (ii) in desmos. Why is this happening?
What does the equation $x^2 = 4$ mean? On its face, it states that we have some $x$ (we'll assume it's a real number), such that $x * x = 4$.
There are two such real numbers. Clearly, $2 * 2 = 4$. But also, $(-2) * (-2) = 4$. So, the equation $x^2 = 4$ has two solutions.
This is true regardless of how we define the square-root function. Since we tend to require that functions map each input to only one output, the canonical "square-root function" takes only non-negative values. But this is merely a convention; one consequence of this convention is that we must use the fairly cumbersome $\pm$ symbol to indicate the presence of both solutions.