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

Summary

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ 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 $x^n-px+q=0$ for arbitrary $n\ge 2$. Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. 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.

Notes

1 We use the form $x^2-px+q=0$ instead of $x^2+px+q=0$ only for convenience; this decision simplifies most calculations in what follows.

2 Note that we hereby obtained one part of Vieta’s formula for a quadratic. We will investigate this later in more detail.

3 By envelope we mean a differentiable function e such that for every p there is one unique x for which Q_x is a tangent to e at the point $(p,e(p))$.

4 By a simple substitution of the form $x\to x+c$ every cubic equation can be transformed into one without quadratic term. For more details on this and various considerations on cubic equations from a mathematics-educational perspective see [2].

5 Recall Vieta’s formula for quadratic equations: u and v are the solutions of the quadratic equation $x^2+ax+b=0$ if and only if $a=-(u+v)$ and b = uv.

6 It is a convention to take the negative.

7 We only needed to write x₀ instead of x to derive the formula for $f^{*}$. For simplicity we omit the index now.

Additional information

Notes on contributors

Michael Schmitz

Michael Schmitz ([email protected]) is a lecturer in mathematics at the University of Flensburg, where he received his Ph.D. in 2014. At the small university in the northernmost city of Germany he is busy educating future primary and secondary mathematics teachers. Together with befriended mathematicians from several countries he regularly organizes and runs math camps for school kids in different locations. When he is not doing or teaching mathematics, he likes traveling, long distance running, reading and spending time with his four girls (wife and three daughters).

André Streicher

André Streicher ([email protected]) is a mathematics and physics teacher and the deputy headmaster of the Klaus-Groth-Schule Neumünster in Northern Germany. He particularly promotes gifted pupils, who are interested in different topics of mathematics and natural sciences. In his free time he likes to keep himself busy with reading, playing the piano, soccer and enjoying time with his family (wife and two daughters).