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How to Solve Cubic Equations

Updated: Aug 2, 2023

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, it is widely taught in middle or high 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 manipulation of the cubic formula requires not only the knowledge of simple algebra, but also that of complex algebra and geometry. Thus, without a proper grasp of the intricacies of them, it is quite unlikely to be able to extract anything out of the cubic formula which has been known to us since mid 1500s. In this review article, we present the derivation of the quadratic and cubic formulae by using the geometrical symmetries in each case. 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.

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