I'm having some trouble understanding this question
Find all solutions in Z16 to the equation x^2 = 9
wouldn't the answer still be 3 or -3 as they're within Z16?
There could be more though. Fortunately by parity argument you really only need to check the odd ones, also, by your observation, the negative one will work too, so really just the odds up to 7 inclusive. I can see by inspection 5 and 11 will also work. So 3,5,11,13 appear to be the solutions mod 16.