I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, techniqu

I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, technique and mental models are left out of modern mathematics education and writing. It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equation-and-beyond https://alexcbecker.net/mathematics.html#the-quadratic-equat...