Is it "$Rightarrow$" or "$Leftarrow$" in "$x^2 = 9 text{ ? } x = 3$"?

$x^2 = 9 iff x^2 - 9 = 0$ is clear. Of course $x^2 -9 = (x-3)(x+3)$, and in all fields (like $Bbb R$ or $Bbb C$) we have that $$ab= 0 iff (a = 0) lor (b=0)tag{1}$$ for all $a,b$ in that field.

So $$x^2 = 9 iff (x-3)(x+3)= 0 iff (x-3 = 0) lor (x+3=0) iff x=3 lor x= -3$$

So every step is reversible by standard facts about inverses and $(1)$.

To begin with, $x^2=9 Leftarrow x=3$ is certainly correct.

Regarding the other way around, $x^2=9 Rightarrow x=3$ it depends. If $x$ earlier has been restricted to be non-negative, e.g. if $x in mathbb{N}$, then the implication is correct. If $x$ can be negative, the implication is not correct, since also $x=-3$ is a possibility.

$$x^2=9Rightarrow x=3$$ is wrong.

Try $x=-3.$ $$x=3Rightarrow x^2=9$$ is true of course.

Also $x^2=9$ is equivalent to $x=3$ or $x=-3$ because $x^2-9=(x-3)(x+3).$