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(1) Euclid's Geometric Solution of Quadratic Equations

 Use the checkboxes (at the left or right) to select one of the two cases of q. Checking the box will mark certain values of p (right axis) - p is a multiple of the raster size, and - the roots x1 and x2 are multiples of the raster size. Select the raster size, or a continuous mode ("Raster off"). A table of p, q, x1, x2 is available by "Data Window".

 The Greeks created a geometric algebra: terms of equations were represented by sides of geometric objects, usually lines, and found by constructive methods. The method in my applets is presented in the paper of Hornsby. It is based on Euclid's Elements, Book II, Proposition 11, and Book VI, Propositions 28 and 29. Using the Right Triangle Altitude Theorem (Altitude-On-Hypotenuse Theorem): q·1 = h2  and x1·x2 = h2 x1·x2 = q and  x1·x2 = |p|  (Vieta)

 Web Links Euclid's Elements Lill's method (Wikipedia) 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) Thomas Carlyle (MacTutor) 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 17