; General cubic (3rd degree polynomial) formula using 3 equations. ; Formula for the 3 roots (solutions for x) of the general cubic equation. ; These formulas always seem to work correctly, whether imaginary or real solutions. ; ; This is currently the only way to solve cubic polynomials in Mathomatic, ; by manually entering the coefficients (a, b, c, and d) into the following equations. ; To visualize the coefficients of an equation: ; solve for 0 ; unfactor ; factor x ; ; See also file "cubic2.in". a x^3 + b x^2 + c x + d = 0 ; The general cubic equation. x_1=-b/{3 a}-1/{3 a} {{2 b^3-9 a b c+27 a^2 d+{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/{2}}^(1/3)-1/{3 a} {{2 b^3-9 a b c+27 a^2 d-{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/2}^(1/3) x_2=-b/{3 a}+{1+i 3^.5}/{6 a} {{2 b^3-9 a b c+27 a^2 d+{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/{2}}^(1/3)+{1-i 3^.5}/{6 a} {{2 b^3-9 a b c+27 a^2 d-{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/2}^(1/3) x_3=-b/{3 a}+{1-i 3^.5}/{6 a} {{2 b^3-9 a b c+27 a^2 d+{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/{2}}^(1/3)+{1+i 3^.5}/{6 a} {{2 b^3-9 a b c+27 a^2 d-{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}^.5}/2}^(1/3) ; x_1, x_2, and x_3 are the solutions to the given general cubic equation. ; Type "calculate all" to temporarily plug in coefficients.