depressedCubic.m

% 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