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# XerxesZorgon/depressedCubic.m

Created Apr 14, 2022
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 % Solves the depressed cubic equation x^3 + Ax + B = 0 % Ref: http://www.sosmath.com/algebra/factor/fac11/fac11.html % % Input(s) % A,B: Coefficients of depressed cubic % % Output(s) % x: Solution to the equation % % Example: % depressedCubic(6,20) % ans = 2 % Check: 2^3 + 6*2 = 20 % % See also: % % % Dependencies: None % % % Written by: John Peach 10-Apr-2022 % Wild Peaches % % Revisions: function x = depressedCubic(A,B) % Equation is of the form x^3 + Ax = B % Solution is x = s - t where % 3st = A % s^3 - t^3 = B % Substitute t = A/(3s) in second equation to get % s^3 - A^3/(27s^3) = B % Let u = s^3, and solve quadratic % u^2 - Bu - A^3/27 = 0 D = sqrt(B^2+4*A^3/27); u = [B + D; B - D]/2; % Substitute back into s s = u.^(1/3); % Keep only the real, nonzero roots if ~isreal(s(2)) || abs(s(2)) < sqrt(eps) s(2) = []; end; if ~isreal(s(1)) || abs(s(1)) < sqrt(eps) s(1) = []; end; % Solve for x t = A./(3*s); x = s - t; endfunction