Accurate algebraic formula for the quintic & Solution by iteration of radicals

TL;DR

This paper introduces a novel iterative radical method for solving the quintic equation, providing an accurate algebraic formula for root approximation and a geometric approach to derive all roots.

Contribution

It presents a new iterative radical solution for the quintic and an explicit algebraic formula with high accuracy for root approximation.

Findings

Root approximation error less than 0.00432

Relative root approximation error less than 0.0251

Develops a geometric trigonometric algorithm for all roots

Abstract

According to the Abel-Ruffini theorem [1] and Galois theory [2], there is no solution in finite radicals to the general quintic equation. This article takes a different approach and proposes a new method to solve the quintic by iteration of radicals. But, the most intriguing result is an accurate algebraic formula for absolute and relative root approximation: |formula - root| < 0.00432 and |formula/root - 1| < 0.0251. We then expand some of the geometric properties discussed to construct a trigonometric algorithm that derives all roots.