Comparative analysis of solutions to $3^{rd}$ and $4^{th}$ order algebraic equations

Three methods of solving a cubic equation are analyzed: del Ferro–Tartaglia’s, Vieta’s, and F. Klein’s. Much attention is given to the irreducible case. Three methods of solving a quartic equation are compared: Ferrari–Cardano’s, Descartes’, and Euler’s. A new derivation of Euler’s formulas for the roots of a quartic algebraic equation is given. Some new methods of symbolic computation of roots of cubic and quartic equations are proposed.