How to solve a cubic equation, part 1: The shape of the discriminant

The approaches for the solution of cubic applications are discussed. The problem associated with the cubic equations is to find the roots x of the equation. The roots consists of [x,w] pairs, where any scaler multiple represents the same root. There are four probabilities with cubic polynomial roots, including- a triple root, a double root and a single root, three distict real roots, or one real and one complex pairs of roots, consisting various permutations of multiple roots and imaginary roots. Computer graphics is used to project homogenous 4D space ito a 3D space including a plane at infinity.