Number of real solutions of quadratic equation

Question

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!

Answer 1

HINT: $$ax^2+bxy+cy^2=0$$ $$c(y^2+\frac{b}{c}xy+\frac{a}{c}x^2)=0$$ $$c(y+m_1x)(y+m_2x)=0$$ where $m_1+m_2=-\frac{b}{c}$ and $m_1m_2=\frac{a}{c}$

Hope this helps.