Now this is the beautiful part:
which is GREAT, because it means our curve secretly has genus 0 and we can actually NAME ALL THE SOLUTIONS.
is the non-trivial case, so let's get at the others:
Now we can be boring and answer the original question here:
(trivial case) or -- by the quadratic formula --
, and checking against the one solution we already know, we see we should take the negative square root, because
was stipulated. And that's OK, if you're into that thing, but I want to do stereographic projection off of
-- the point I get when I let
so that I can find all solutions to your equation just by plugging in some rational numbers.
and this is MUCH cooler, because you can see the quadratic formula in it again, only this time it's much better because we already know one solution, namely
, and the other is given by solving:
which finally is an easy task:
I personally think this is even better than the original answer because if you put in any rational
you can get all the rational points on the curve, and if you think about it this gives you all integer solutions to your original question. It's also easy to see that you only need rationals so that
, and it's easy to solve for the appropriate intervals. Fun problem all in all.