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/2014/10934Abstract
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.
References (7)
- 1 Time Series Analysis, Forecasting, and Control [DOI]
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- 6 Use of a Cubic Equation to Predict Surface Tension and Spinodal Limits [DOI]
- 7 Roots of polynomials of degrees 3 and 4 [DOI]
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