Interesting Esoterica

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

Article by Michael Schmitz and André Streicher
  • Published in 2020
  • Added on
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.

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BibTeX entry

@article{Envelopesaresolvingmachinesforquadraticsandcubicsandcertainpolynomialsofarbitrarydegree,
	title = {Envelopes are solving machines for quadratics and cubics and certain  polynomials of arbitrary degree},
	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.},
	url = {http://arxiv.org/abs/2012.06821v1 http://arxiv.org/pdf/2012.06821v1},
	year = 2020,
	author = {Michael Schmitz and Andr{\'{e}} Streicher},
	comment = {},
	urldate = {2021-01-06},
	archivePrefix = {arXiv},
	eprint = {2012.06821},
	primaryClass = {math.HO},
	collections = {easily-explained,fun-maths-facts,geometry,things-to-make-and-do,unusual-computers}
}