Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, each of which determines a field of directions on $x(U)\subset S$. Prove that these two fields of directions are othorgonal if and only if $$Ec-2Fb+Ga=0$$ (where $x_{u}x_{u}=E,x_{v}x_{u}=F$ and $x_{v}x_{v}=G)$.
I am confused about this question, I don't know how to factor the first equation and what is the relation between two equations? Can someone help me solve this question?