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!