To find the point between two massive spherical bodies at which their gravitational fields cancel is an apparently simple problem usually found in introductory physics textbooks. However, by noting that such a point does not exist when the distance between the spheres is small and one of the masses is much smaller than the other--e.g., between the Earth and a billiard ball near the ground--I develop here a simple analysis for establishing existence conditions for this point. Unexpectedly, I have found that the net gravitational field can be null also in certain points inside each sphere. The position of these "inner" points can be determined by solving a cubic equation via the standard method, known as "Cardan's solution". However, when the discriminant of this equation is negative one has the "irreducible" case, for which Cardan's solution "fails", but a trigonometric method proposed recently yields exact closed-form expressions. Interestingly, it is shown that these "inner points" do occur in the Earth-Moon system, in which they are determined just by solving irreducible cubic equations. (Contains 13 footnotes and 2 figures.)