Solving the quartic by conics - Halbeisen Lorenz; Hungerbühler Norbert (Open Mathematics)
Two conic sections C1C_{1} and C2C_{2} in the Euclidean plane that pass through two given points can generally have two further points of intersection. It is shown how these can be constructed using compass and ruler. The idea is to construct the degenerate conics in the pencil of the two conics C1C_{1} and C2C_{2}. Their intersections are then the four intersection points of C1C_{1} and C2C_{2}. The same idea is then used to reduce a general quartic equation to a cubic equation and to solve it. This is performed by interpreting the solutions of the quartic as intersections of two complex conic sections. The degenerate complex conics in their pencil can then be found through a cubic equation.