x=3 <=> x^2 = 9 because x=3 <=> x^2= x * x = 3 * 3 = 9

x^2 = 9 is not <=> x=3 because (x^2 = 9) is a function that accepts two solutions x= 3 or x=-3

x^2 = 9 is <=> x^2 - 3^2 = 0 is <=> (x-3)*(x+3)=0 is <=> x-3=0 or x+3=0

is <=> x=3 or x=-3. This is the correct way.

So x=3 => x^2 = 3^2. The sol of the left is {3}; the sol of the right is {-3,3}. {3} is a subset of {-3,3}. So x=3 => x^2=9 ( not <=> ).