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(4) Kumar's Geometric Solution of Quadratic Equations

The only tools required by Kumar's technique are a sheet of graph paper, and a setsquare (or a 45° right triangle).

Kumar geometric solution
                  of quadratic equation

x2 + px + q = 0

- Plot the point (q,p).

- Place the setsquare ABC:
     with its edge AC passing through the point (1,0) on the horizontal axis,
     with its apex A on the vertical axis,
     with the perpendicular edge AB passing through the point (q,p).

- The ordinate of the point A on the vertical axis gives the negative value of one of the roots.

Quadratic equation: x2 + px + q = 0
Example: p=3, q=2,
roots: x1=-1, x2=-2

The proof is given in the paper of Arun Kumar.
raster box
Checking the box will mark certain values q, and p:
-  q, and p are a multiple of the raster size, and
-  the roots x1 and x2 (if any) are multiples of the raster
Select the raster size, or a continuous mode ("Raster off").
A table of p, q, x1, x2 is available by "Data Window".
The curve p = 2·sqrt(q) represents the limit between the regions of (p,q) with reals roots existing and no real roots. It corresponds to the discriminant D = sqrt(p2/4 - q) = 0.

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).

(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.

Arun Kumar: A new technique of solving quadratic equations, Journal of Recreational Mathematics, Vol 14(4), 1981-82, pp 266-270.

Updated: 2023, Oct 06