(1-x-1/x+1)(x+2)=x+1/x-1+x-1/x+1

3 min read Jun 16, 2024

Solving the Equation: (1-x-1/x+1)(x+2)=x+1/x-1+x-1/x+1

This problem involves solving an equation with fractions and parentheses. To solve it effectively, we need to simplify the equation step by step. Let's break down the process:

1. Simplifying the Left-Hand Side

Combine terms inside the parentheses: (1-x-1/x+1) can be written as (x-x^2-1)/(x+1).
Multiply by (x+2): [(x-x^2-1)/(x+1)] * (x+2) = (x-x^2-1)(x+2)/(x+1)

2. Simplifying the Right-Hand Side

Find a common denominator: The common denominator for (x+1)/(x-1) and (x-1)/(x+1) is (x-1)(x+1).
- (x+1)/(x-1) becomes (x+1)(x+1)/(x-1)(x+1)
- (x-1)/(x+1) becomes (x-1)(x-1)/(x-1)(x+1)
Combine the terms: [(x+1)(x+1) + (x-1)(x-1)]/(x-1)(x+1) = (x^2 + 2x + 1 + x^2 - 2x + 1)/(x-1)(x+1) = (2x^2 + 2)/(x-1)(x+1)

3. Combining the Simplified Sides

Now our equation looks like this: (x-x^2-1)(x+2)/(x+1) = (2x^2 + 2)/(x-1)(x+1)

4. Solving for x

Multiply both sides by (x+1)(x-1): (x-x^2-1)(x+2)(x-1) = 2x^2 + 2
Expand and simplify: -x^3 + 3x^2 - 3x + 2 = 2x^2 + 2
Rearrange to a standard form: x^3 - x^2 + 3x = 0
Factor out x: x(x^2 - x + 3) = 0
Solve for x: x = 0 or x^2 - x + 3 = 0

The quadratic equation x^2 - x + 3 = 0 does not have real solutions.

5. Solution

Therefore, the only solution to the equation is x = 0.