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Research Article Open access CC BY 3.0

Some Contributions to the Solution of Cubic Equations

O. E. Okereke, I. S. Iwueze, J. Ohakwe

Journal of Advances in Mathematics and Computer Science · pp. 2929–2941 · Published 7 Aug 2014

10.9734/BJMCS/2014/10934

Abstract

In many applications, the solution of a cubic equation is required. This study improves on the existing methods of solving cubic equations. The general expression for the quadratic factor of a cubic equation is derived. The discriminant of the cubic equation, adopted in this study as D, comprises of two terms, D1 and D2 such that D = D12 - D23 and the value of D is shown to depend on the nature of the root of a given cubic equation. For a cubic equation to have three distinct real roots, D<0. If  D>0, the equation has only one real root, and if D = 0, the equation has either two equal real roots or three equal real roots. A required and sufficient condition for a cubic equation to have three equal real roots is shown to be D1 = 0, D2 = 0 and D = 0. It is also established that a cubic equation has two real equal roots if D1 ≠ 0, D2 ≠ 0, but D12 =  D23 and D = 0. The solution of a cubic equation with three real equal roots is found using a method that depends on the corresponding reduced cubic equation. Finally, numerical examples are used to demonstrate the established results.

Cubic equation discriminant quadratic factor root cubic characteristic equations.

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