Solving the Equation (frac{1}{3x^2-3x}-frac{1}{2x^2-2x}1): A Comprehensive Guide

Introduction

When faced with complex algebraic equations, understanding the step-by-step process can be challenging. In this article, we will guide you through solving the equation (frac{1}{3x^2-3x}-frac{1}{2x^2-2x}1). We will break it down to make it understandable and ensure you grasp the methodology behind solving such equations.

Understanding the Equation

The equation in question is:

(frac{1}{3x^2-3x}-frac{1}{2x^2-2x}1)

At a glance, this equation can appear daunting. However, by breaking it down into simpler steps, we can solve it effectively.

Step 1: Simplify the Denominators

The first step in solving the equation is to simplify the terms in the denominators. Notice that both terms in the denominators share common factors:

(3x^2-3x 3x(x-1))

(2x^2-2x 2x(x-1))

Using these simplifications, the equation becomes:

(frac{1}{3x(x-1)} - frac{1}{2x(x-1)} 1)

Step 2: Combine the Fractions

The next step is to combine the fractions. Notice that both fractions have a common denominator:

(frac{2 - 3}{6x(x-1)} 1)

Simplifying the numerator:

(frac{-1}{6x(x-1)} 1)

Step 3: Eliminate the Denominator

To eliminate the denominator, multiply both sides of the equation by (6x(x-1)):

(-1 6x(x-1))

Expanding the right-hand side:

(-1 6x^2 - 6x)

Step 4: Rearrange the Equation

Move all terms to one side of the equation to set it to zero:

(6x^2 - 6x 1 0)

Step 5: Solve the Quadratic Equation

The equation now is in the form of a standard quadratic equation (ax^2 bx c 0), where (a6), (b-6), and (c1). To solve this quadratic equation, we can use the quadratic formula:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

Substituting the values of (a), (b), and (c):

(x frac{-(-6) pm sqrt{(-6)^2 - 4(6)(1)}}{2(6)})

(x frac{6 pm sqrt{36 - 24}}{12})

(x frac{6 pm sqrt{12}}{12})

(x frac{6 pm 2sqrt{3}}{12})

(x frac{3 pm sqrt{3}}{6})

(x frac{3 pm sqrt{3}}{6})

Simplifying further:

(x frac{1}{2} pm frac{sqrt{3}}{6})

The solutions are:

(x_1 frac{1 sqrt{3}}{6})

(x_2 frac{1 - sqrt{3}}{6})

Conclusion

By following these steps, we have successfully solved the given algebraic equation. The solutions to the equation are (x frac{1 sqrt{3}}{6}) and (x frac{1 - sqrt{3}}{6}).

Understanding the process of breaking down a complex equation into simpler steps is crucial for solving various mathematical problems. If you have any questions or comments, please leave them below!