Simple Formulae for the Evaluation of All the Exact Roots (Real and Complex) of the General Cubic

  • AJ Anyaegunam
Keywords: General cubic, trigonometric, hyperbolic, simple formulae, complex roots


Simplified explicit formulae for the evaluation of the exact roots of the general cubic equation ax3 + bx2 + cx + d  =  0 have been formulated and presented. The explicit hyperbolic expressions for the complex roots have been developed for the first time in history thereby enabling the establishment of harmony in the solution of cubic equations. Also, four alternative expressions for the only real root of the cubic have also been given for those cases when the discriminant is positive.

Research papers

Journal Identifiers

eISSN: 2467-8821
print ISSN: 0331-8443