Does x^2 - 3x + 7 have a root in the integers modulo 73?

Question

anonymous · Answer

Also, how to show it.

anonymous · Answer

Are we trying to show that:
$$x^2-3x+7 \equiv 0 \mod 73$$
is possible (or not)?  Just trying to wrap my head around the question.

anonymous · Answer

Yeah.  The couple of questions before it dealt with quadratic reciprocity, although I don't know for sure that this one does.

anonymous · Answer

I know the roots are 24 and 52, but I don't know a "simple" way to show that there is a root.

anonymous · Answer

blah, ive looked through two books i have lying around and im not finding a sorta "faster" way to show it.

anonymous · Answer

Oh well, thanks for trying!

anonymous · Answer

Maybe this would help, its a problem from a section on the Legendre Symbol. Using this, to show that there is a solution, you would just need to compute:
$$\bigg(\frac{-19}{73}\bigg)$$

anonymous · Answer

I love it!  This looks perfect.  I'll take a look this afternoon.

anonymous · Answer

The proof seemed pretty easy.  Thanks for putting that problem up!