Envelopes are solving machines for quadratics and cubics and certain polynomials of arbitrary degree

TL;DR

This paper introduces a geometric method using a single envelope curve to solve quadratic, cubic, and higher-degree polynomial equations, simplifying the solution process and revealing connections to duality and Legendre transformation.

Contribution

It presents a novel envelope-based geometric technique for solving polynomials of arbitrary degree, extending classical methods and linking to duality and Legendre transformation.

Findings

A single envelope curve can solve all quadratic equations graphically.

The method generalizes to polynomials of any degree n ≥ 2.

Immediate visualization of the number of solutions for specific equations.

Abstract

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn. Stunningly, there is one single curve that can be used to solve every quadratic equation via drawing tangent lines through a given point $(p,q)$ to this curve. In this article we derive this method in an elementary way and generalize it to equations of the form $x^n-px+q=0$ for arbitrary $n \ge 2$. Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. Concluding the article we point out connections to the duality of points and lines in the plane and to the the concept of Legendre transformation.