Solving Algebraic Equations

Abstract

The aim of this chapter is to show how equations of degrees less than 5 can be solved. We highlight well-known formulae for the quadratic equation and show how to find similar formulae for cubic and quartic equations. We also explain why as early as the eighteenth century mathematicians started to doubt the possibility to find solutions for general quintic equations (or equations of higher degrees) using the four arithmetic operations and extracting roots applied to coefficients. We give examples of quantic equations for which such formulae exist (e.g. de Moivre’s quintics) and show that the ideas which work for equations of degrees up to 4 have no evident generalizations. We also briefly discuss “casus irreducibilis” related to cubic equations.