Solving the Equation: 1/2 - x - 1 = 1/x - 2 - 6/x - 12/x^2
Introduction
In this article, we will solve the equation 1/2 - x - 1 = 1/x - 2 - 6/x - 12/x^2. This equation involves rational expressions and requires careful manipulation to solve for x.
Step-by-Step Solution
First, let's start by combining the fractions on the left-hand side of the equation:
1/2 - x - 1 = (1 - 2x - 2) / 2
Next, let's combine the fractions on the right-hand side of the equation:
1/x - 2 - 6/x - 12/x^2 = (x - 2 - 6 - 12/x) / x^2
Now, we can set the two expressions equal to each other:
(1 - 2x - 2) / 2 = (x - 2 - 6 - 12/x) / x^2
To simplify the equation, we can multiply both sides by 2x^2 to eliminate the fractions:
x^2 - 2x^2 - 2x - 4x^2 = x - 2 - 6 - 12
Now, let's rearrange the terms to group like terms together:
-4x^2 - 2x - 2 = -x - 18
Next, we can add x to both sides of the equation:
-4x^2 - x - 2 = -18
Now, we can add 18 to both sides of the equation:
-4x^2 - x + 16 = 0
This is a quadratic equation in x. We can factor the left-hand side:
(-2x + 4)(2x - 4) = 0
This gives us two possible solutions for x:
x = 2 or x = 2/2 = 1
Therefore, the solutions to the equation are x = 2 and x = 1.
Conclusion
In this article, we have solved the equation 1/2 - x - 1 = 1/x - 2 - 6/x - 12/x^2 using a combination of fraction operations and quadratic equation factorization. The solutions to the equation are x = 2 and x = 1.
