Discrete Problem Help

Hey all,

I need help on this problem. Say that x^2 is congruent to 3 (mod 83).
Find the inverse of x (mod 83).

I made a small example case first: (I think all this is wrong now) Please help!

x^2-3 divides 83...so if you make x=3 into x^2-3 you get 9-3=6

So then you do 6^-1 (mod 83) //Reads as 6 inverse 83

So I did this next:

83=13*6+5
6=1*5+1

Working backwards
1=6-1*5
=6-1(83-13*6)
=14*6-1*83

So in conclusion 14 is the answer to 6 inverse 83.

What's confusing is, is this the answer to inverse of x(mod 83)?? I think its wrong, am I being misled, please help me.

Thanks to those who help.

-Coder752