Envelopes are Solving Machines for Quadratics and Cubics and Certain Polynomials of Arbitrary Degree

Abstract

Summary Everybody knows from school how to solve a quadratic equation of the form x 2 − p x + q = 0 graphically. To solve more than one equation this method can become tedious, 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 by 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 − p x + q = 0 for arbitrary n ≥ 2 . Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. In concluding the article, we point out connections to the duality of points and lines in the plane and to the concept of Legendre transformation.