Last update: 30 June 2012
We wish to solve Divide by to get Since So So Replacing and gives So the two solutions for are
We wish to solve Put . Then and so we may assume that our original equation was of the form Let Then implies that So Thus and so So and This is the solution of by radicals.
There are solutions by radicals of quadratic, cubic and fourth degree polynomials. Abel and Galois proved that the general solutions of a degree 5 polynomial in is not given by radicals. This page gives derivations of the solutions by radicals for quadratic and cubic polynomials.
The presentation of the qudratic formula was handwritten in 2012 to fill out this page which already had the derivation of the cubic formula. The derivation of the cubic formula is from the "Group cohomology" page of Work2004/BookNewalg/PartV.pdf (page 20). No reference for the source of this derivation is in the original notes.