Solving the Equation m^3 - m^2 36: A Step-by-Step Guide

Introduction

Algebraic equations, especially cubic equations, can often be challenging to solve. In this guide, we will walk you through the process of solving the equation m^3 - m^2 36. Understanding how to solve such equations is crucial for various fields of study, including engineering, physics, and mathematics.

Step 1: Rearrange the Equation

To begin, we need to rearrange the given equation to a standard form. The equation is:

m^3 - m^2 - 36 0

Factorizing the Polynomial

Let's define a polynomial function f(m) m^3 - m^2 - 36. We can test for possible integer roots by substituting values into f(m).

Test for Integer Roots

By substituting m -3 into the polynomial:

f(-3) (-3)^3 - (-3)^2 - 36 -27 - 9 - 36 -72 0 0

This indicates that m -3 is a root of the polynomial. Therefore, (m 3) is a factor of m^3 - m^2 - 36.

Step 2: Polynomial Division

Now, we can use polynomial division to find the remaining quadratic factor:

m^3 - m^2 - 36 (m 3)(m^2 - 4m - 12)

Next, we need to solve the quadratic equation m^2 - 4m - 12 0.

Quadratic Formula for the Quadratic Equation

The quadratic formula is given by:

m frac{-b pm sqrt{b^2 - 4ac}}{2a}

For the quadratic m^2 - 4m - 12, we have a 1, b -4, and c -12.

m frac{4 pm sqrt{(-4)^2 - 4 cdot 1 cdot (-12)}}{2 cdot 1} frac{4 pm sqrt{16 48}}{2} frac{4 pm sqrt{64}}{2} frac{4 pm 8}{2}

This gives us two solutions:

m frac{4 8}{2} 6
m frac{4 - 8}{2} -2

Step 3: Conclusion

Therefore, the solutions to the equation m^3 - m^2 36 are:

m -3, m 6, and m -2

Graphical Interpretation

Graphically, we can visualize the equation m^3 - m^2 - 36 0 as the intersection of the curve y m^3 - m^2 - 36 with the x-axis. The roots correspond to the x-coordinates where the curve intersects the x-axis.

Conclusion

By following these steps, you can solve the equation m^3 - m^2 36 and find its roots. This process involves both algebraic methods and the application of the quadratic formula, providing a comprehensive understanding of solving cubic equations.