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How to Solve Cubic Equations

Updated: Aug 2, 2023

by ARIF ERKOCA, PhD



While solving most of the science and engineering problems, we frequently encounter the need for the use of the formula which gives the solutions to a quadratic equation, and is taught widely in middle schools all around the world. However, the formula for the solutions to a cubic equation is never mentioned, not even at higher levels of education. This is mainly due to the fact that the utilization of the cubic formula requires not only the knowledge of algebra, but also that of complex algebra and trigonometry. Thus, without a proper grasp of the intricacies of these topics, it is quite unlikely to be able to extract anything out of the cubic formula which has been known to us since mid 1500s. Furthermore, a brief knowledge on analytical geometry and calculus also provides a better understanding of the derivation of the cubic formula. In this review article, we present the derivation of the quadratic and cubic formulae by using the symmetry properties of the polynomial functions. We put special emphasis on obtaining the conditions which yield the number of real solutions to a cubic equation. Since the use of complex algebra is inevitable even when a cubic equation has three real solutions, we give a concise review of complex algebra in the Appendix. Trigonometry also plays a major role in obtaining the solutions to cubic equations with real solutions. We also present a detailed study on how to solve cubic equations with the use of trigonometry.

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