Solving Cubic Equations with Triangles: A Geometric Approach to Root Selection
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Abstract
In many practical situations, engineers encounter problems that lead to quadratic equations. These equations have two possible solutions, and it is usually straightforward to determine which one is relevant using physical principles. However, some problems lead to cubic equations, which are more complex. While some cubic equations have one real root and two complex conjugate roots, others possess three distinct real solutions. This raises a natural and important question: how can we decide which of these roots corresponds to the physical reality of the problem?. In this article, a geometric tool—the equilateral triangle method—is presented to help identify the physically meaningful solution of a cubic equation. This method is illustrated through three engineering problems that arose during the course of the author's research: (1) the derivation of a kinetic equation for the removal of magnesium from liquid aluminum, (2) the analysis of the behavior of a magnetocaloric material, and (3) the determination of efficiency in a wind turbine blade.
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