Solving the Equation: 5/x - 3 + 4/x + 3 = x - 5/x^2 - 9
In this article, we will discuss the solution to the equation 5/x - 3 + 4/x + 3 = x - 5/x^2 - 9. This equation involves fractions and square operations, making it a bit challenging to solve. However, with the right approach, we can simplify the equation and find the solution.
Step 1: Combine Like Terms
First, let's combine the like terms on both sides of the equation:
5/x - 3 + 4/x + 3 = x - 5/x^2 - 9
Combine the fractions on the left side:
(5 + 4)/x - 3 + 3 = x - 5/x^2 - 9
Simplify the equation:
9/x = x - 5/x^2 - 9 + 0
Step 2: Multiply Both Sides by x^2
To eliminate the fractions, multiply both sides of the equation by x^2:
9x = x^3 - 5x - 9x^2
Step 3: Rearrange the Terms
Rearrange the terms to make it easier to solve:
x^3 - 9x^2 - 5x - 9 = 0
Step 4: Factorize the Equation
Factorize the equation to find the solutions:
(x - 9)(x + 1)(x + 1) = 0
Step 5: Solve for x
Solve for x by setting each factor equal to 0:
x - 9 = 0 --> x = 9
x + 1 = 0 --> x = -1 (twice)
Therefore, the solutions to the equation are x = 9 and x = -1.
Conclusion
In conclusion, we have successfully solved the equation 5/x - 3 + 4/x + 3 = x - 5/x^2 - 9 by combining like terms, multiplying both sides by x^2, rearranging the terms, factorizing the equation, and solving for x. The solutions to the equation are x = 9 and x = -1.
