1/3-x-1/x+1=x/x-3-(x-1)^2/x^2-2x-3

3 min read Jun 16, 2024

Solving the Equation: 1/3 - x - 1/x + 1 = x/x - 3 - (x-1)^2/x^2 - 2x - 3

In this article, we will solve the equation 1/3 - x - 1/x + 1 = x/x - 3 - (x-1)^2/x^2 - 2x - 3. This equation may seem complex, but by using algebraic manipulations and simplifications, we can solve for x.

Step 1: Simplify the Equation

First, let's simplify the equation by combining like terms:

1/3 - x - 1/x + 1 = x/x - 3 - (x-1)^2/x^2 - 2x - 3

We can start by combining the fractions on the left-hand side:

(3 - 3x - 1 + 3)/3x = x/x - 3 - (x-1)^2/x^2 - 2x - 3

Simplifying the numerator, we get:

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

Step 2: Multiply Both Sides by 3x

To eliminate the fraction on the left-hand side, we multiply both sides of the equation by 3x:

1 - 3x = 3x^2 - 9x - (x-1)^2 - 6x^2 - 9x

Step 3: Expand and Simplify

Expanding the right-hand side of the equation, we get:

1 - 3x = 3x^2 - 9x - (x^2 - 2x + 1) - 6x^2 - 9x

Combine like terms:

1 - 3x = -4x^2 + 2x - 1

Step 4: Rearrange the Equation

Rearranging the equation to make x the subject, we get:

4x^2 + x - 2 = 0

Step 5: Factorize the Quadratic Equation

Factoring the quadratic equation, we get:

(2x + 1)(2x - 2) = 0

Step 6: Solve for x

Solving for x, we get:

2x + 1 = 0 or 2x - 2 = 0

x = -1/2 or x = 1

Therefore, the solutions to the equation are x = -1/2 and x = 1.

In conclusion, we have successfully solved the equation 1/3 - x - 1/x + 1 = x/x - 3 - (x-1)^2/x^2 - 2x - 3, and the solutions are x = -1/2 and x = 1.