Why is the solution to $x-\sqrt 4=0$ not $x=\pm 2$? If the equation is $x-\sqrt 4=0$, then $x=2$.
If the equation is $x^2-4=0$, then $x=\pm 2$.
Why is it not $x=\pm 2$ in the first equation?

$x^2 - 2^2 = 0$
$(x+2)(x-2)=0$
$x+2=0$ or $x-2=0$
$\sqrt4 = 2$
$x-\sqrt4=0$
$x=2$
Why can't the square root of $4$ be $-2$ instead of $2$, if $-2$ times $-2$ also equals $4$?
