I have the following question that puzzles me: How do I determine the number of non-trivial real solutions to the general equation $ax^2 + bxy + cy^2 = 0$ (up to a scalar)?
My attempt was to fix $y \neq 0$ and then see how many non-complex solution the quadratic equation has for $x$. However, I was hoping that there would be a more elegant way to do this.
Thanks for any help!