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