Cubic Roots

cubic_roots(first_constant, second_constant, third_constant, fourth_constant, precision=4)

Calculates the roots of a cubic function

Parameters

  • first_constant (int or float) – Coefficient of the cubic term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • second_constant (int or float) – Coefficient of the quadratic term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • third_constant (int or float) – Coefficient of the linear term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • fourth_constant (int or float) – Coefficient of the constant term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the resultant roots

Raises

  • TypeError – First four arguments must be integers or floats

  • ValueError – Last argument must be a positive integer

Returns

roots – List of the x-coordinates of all of the x-intercepts of the original function

Return type

list of float

See also

cubic_equation(), cubic_derivatives(), cubic_integral(), cubic_model()

Notes

  • Standard form of a cubic function: \(f(x) = a\cdot{x^3} + b\cdot{x^2} + c\cdot{x} + d\)

  • Cubic formula: \(x_k = -\frac{1}{3a}\cdot(b + \xi^k\cdot{\eta} + \frac{\Delta_0}{\xi^k\cdot{\eta}})\)

    • \(\Delta_0 = b^2 - 3ac\)

    • \(\Delta_1 = 2b^3 - 9abc +27a^2d\)

    • \(\xi = \frac{-1 + \sqrt{-3}}{2}\)

    • \(\eta = \sqrt[3]{\frac{\Delta_1 \pm \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2}}\)

    • \(k \in \{ 0, 1, 2 \}\)

  • Cubic Formula

Examples

Import cubic_roots function from regressions library
>>> from regressions.analyses.roots.cubic import cubic_roots
Calculate the roots of a cubic function with coefficients 2, 3, 5, and 7
>>> roots_first = cubic_roots(2, 3, 5, 7)
>>> print(roots_first)
[-1.4455]
Calculate the roots of a cubic function with coefficients 7, -5, -3, and 2
>>> roots_second = cubic_roots(7, -5, -3, 2)
>>> print(roots_second)
[-0.6431, 0.551, 0.8064]
Calculate the roots of a cubic function with all inputs set to 0
>>> roots_zeroes = cubic_roots(0, 0, 0, 0)
>>> print(roots_zeroes)
[-1.0]