Suppose we have a second-order curve in general form:

(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.

I'd like to know if there is a simple condition that ensures that the curve has at least one point on on or below the $x$ axis, *i.e.* that the left-hand side of (1) is nonpositive.

In the trivial case that the curve is a parabola, the discriminant being nonnegative is just such a condition. But what happens in the general case?