Geometric Solutions of Quadratic Equations

1. Euclid's Method




Quadratic equation: x² + px + q = 0

Example: coefficients p=4.5, q=2, solved by the roots x1=-0.5, x2=-4

Applet


2. Carlyle's Method




Quadratic equation: x² + px + q = 0

Example: coefficients p=2.5, q=-1.5 solved by the roots x1=0.5, x2=-3

Applet

3. Von Staudt's Method



Applet


4. Kumar's Method




Quadratic equation: x² + px + q = 0

Example: coefficients p=3, q=2 solved by the roots x1=-1, x2=-2

Applet

Web Links

Lill's method (Wikipedia) 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) Felix Klein: Famous Problems of Elementary Geometry, 1941, p. 34f (Google Books)

Print

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. E. John Hornsby: Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal, 1990, Volume 21, Number 5, p. 362-369. Arun Kumar: A new technique of solving quadratic equations, Journal of Recreational Mathematics, Vol 14(4), 1981-82, pp 266-270. Howard Eves: An Introduction to the History of Mathematics, Saunders College Publishing, 6th ed. 1990, Chapter 3-7 (pp 87-90, 99).

Updated: 2012, Apr 10