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Overview

(3) Von Staudt's Geometric Solution of Quadratic Equations







Karl Georg Christian von Staudt describes a method like this in a paper of 1841.

More details can be found in a publication of Felix Klein:

Felix Klein
raster box

euclid
                raster
The option is available if 4/p has been fixed (upper blue axis). Checking the box will mark certain values of q/p (lower red axis)
- q is a multiple of the raster size, and
- the roots x1 and x2 are multiples of the raster size.

quadratic
                equation
Select the raster size, or a continuous mode ("Raster off").
A table of p, q, x1, x2 is available by "Data Window".

von staudt
                  roots of quadratic equations

Quadratic equation: x2 + px + q = 0
Example: 4/p=2, q/p=4, equivalent p=2, q=-8,
roots: x1=-4, x2=2




Web Links

v. Staudt: Construction des regulären Siebzehnecks, Journal für die reine und angewandte Mathematik, 24. Band, Heft 3, 1842, p. 251 (Göttinger Digitalisierungszentrum)

Karl Georg Christian von Staudt (Wikipedia)

Felix Klein: Vorträge über ausgewählte Fragen der Elementargeometrie, 1895, p. 27 ff (Google Books)

Felix Klein: Famous Problems of Elementary Geometry, 1941, p. 34f (Google Books)

Karl Georg Christian von Staudt (MacTutor)

Geometric Construction of Roots of Quadratic Equation (Cut The Knot)

Eduard Lill, radici immaginarie di un polinomio

Der Kreis von Lill, in: R. Kaendes, R. Schmidt (Hrsg.): Mit GeoGebra mehr Mathematik verstehen (Google Books)

Carlyle Circle (Wolfram MathWorld)

Carlyle Circle (Wolfram Demonstrations Project)

Applet showing Lill's method applied to quadratic equations

D. Tournès: Constructions d'équation algébriques et différentielles

T. C. Hull: Solving Cubics With Creases: The Work of Beloch and Lill (PDF)

M. E. Lill: Résolution Graphique des équations numériques de tous les degrées à une seule inconnue, et description d'un instrument inventé dans ce but, Nouvelles Annales de Mathematiques, Series 2, Vol. 6, 1867 ( PDF)

D. W. DeTemple: Carlyle Circles and the Lemoine Simplicity of Polygon Constructions (PDF)

Thomas Carlyle (MacTutor)

Burton: The History of Mathematics, McGraw-Hill, 2006, (PDF).

Print
(1) E. John Hornsby: Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal, 1990, Volume 21, Number 5, p. 362-369.

R. Kaendes, R. Schmidt (Hrsg.): Mit GeoGebra mehr Mathematik verstehen, Vieweg+Teubner, 2011, ISBN 978-3-8348-1757-0.

A. Baeger: Eine geometrische Lösung der quadratischen Gleichung x2 + px + q = 0, in: CASIO Forum 1/2012, CASIO Europe.

E. J. Barbeau: Polynomials, Springer New York Heidelberg Berlin 2003, ISBN 0-387-40627-1, 978-0387-406275.





Updated: 2012, Feb 18