Some_Math

 Giving myself a little break, today, with a math problem that showed up on YouTube. 

It's a cubic polynomial, and although I think of myself as being able to handle these, I

found myself... well, a little rusty!

The problem: -k^3 + k^2 -12 = 0

Gemini Ai, to the rescue:

mathematics; help me solve the following cubic equation: -k^3 + k^2 -12 = 0. How many roots does it have, and how would I know?

                                                                                 

Python code for the graph:

import sympy as sp

import numpy as np

import matplotlib.pyplot as plt

k = sp.symbols('k')

equation = -k**3 + k**2 - 12

# Find roots

roots = sp.solve(equation, k)

print(f"Roots: {roots}")

# Calculate discriminant

# For ax^3 + bx^2 + cx + d = 0

a, b, c, d = -1, 1, 0, -12

discriminant = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2

print(f"Discriminant: {discriminant}")

# Plotting the function to show the real root

k_vals = np.linspace(-4, 4, 400)

f_vals = [-val**3 + val**2 - 12 for val in k_vals]

plt.axhline(0, color='black', linewidth=1)

plt.axvline(0, color='black', linewidth=1)

plt.plot(k_vals, f_vals, label='f(k) = -k^3 + k^2 - 12')

plt.scatter([-2], [0], color='red', label='Real root (k=-2)')

plt.title('Plot of f(k) = -k^3 + k^2 - 12')

plt.xlabel('k')

plt.ylabel('f(k)')

plt.legend()

plt.grid(True)

plt.savefig('cubic_plot.png')

plt.show()