Geometric solutions of algebraic equations

The abcd coefficients of a cubic equation are represented by a set of 4 consecutive horizontal-vertical segments shown in red. At each of  the ABC point , a 90o turn  is performed in the positive or anticlockwise direction if the new coefficient is positive, and in the reverse direction for a negative one. In this example, all coeficients are positive. A ray is drawn beginning at the start point S, first reflecting at 90° on the b segment (or its extension), and again at 90° on the c segment. A solution is obtained by sliding point P on the b segment so that the last ray passes through the finish point F. The (real) root is then equal to x = -tan(γ). The new third-order path shown in blue represents the quadratic equation obtained after extracting the first root from the original cubic equation. The real solutions of the quadratic (if they exist) can be obtained by finding the intersections of the semi circle of diameter SF with the b1  segment.

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     1. M. Riaz, "Geometric Solutions of Algebraic Equations'', The American Mathematical Monthly,Vol.69, No.7,(Aug-Sept 1962),654-658.  Press View #1.
2. M. E. Lill, "Résolution graphique des équations numériques de tous les degrés", Nouvelles Annales de Mathématiques, Série 2, vol 6 (1867), 359-362. Press View #2.
     3. W.H. Bixby,"Graphical Solution of Numerical Equations", The American Mathematical Monthly, Vol.29,(Oct 1922),344-346.
     4. P.V. Bradford, "Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths". Press
View #3.


© M. Riaz