Solving the Equation: 1 + 2x - 5/x - 2 - 3x - 5/x - 1 = 0
In this article, we will solve the equation 1 + 2x - 5/x - 2 - 3x - 5/x - 1 = 0. This equation may seem complex at first, but by using algebraic manipulations, we can simplify it and find its solutions.
Step 1: Combine like terms
First, let's combine the constant terms:
1 - 2 - 1 = -2
So, the equation becomes:
2x - 5/x - 3x - 5/x = 2
Step 2: Simplify the fractions
Next, let's simplify the fractions by finding a common denominator, which is x. We can rewrite the equation as:
2x - 5/x - 3x - 5/x = 2
Multiplying both sides by x to eliminate the fractions, we get:
2x^2 - 5 - 3x^2 - 5 = 2x
Step 3: Rearrange the equation
Now, let's rearrange the equation to get all the terms on one side:
x^2 - 2x - 10 = 0
Step 4: Factor the quadratic equation
Factoring the quadratic equation, we get:
(x - 5)(x + 2) = 0
Step 5: Solve for x
Setting each factor equal to 0, we get:
x - 5 = 0 --> x = 5
x + 2 = 0 --> x = -2
Therefore, the solutions to the equation are x = 5 and x = -2.
In conclusion, by applying algebraic manipulations and simplifications, we were able to solve the equation 1 + 2x - 5/x - 2 - 3x - 5/x - 1 = 0 and find its solutions, x = 5 and x = -2.
