-2/x-1=x-8/x+6
To solve the equation -2/x - 1 = x - 8/x + 6, we can start by clearing the denominators.
To do this, we can multiply every term in the equation by x(x+6) to eliminate the fractions. This will give us:
(x(x+6))*(-2/x) + (x(x+6))*(-1) = (x(x+6))*(x) - (x(x+6))*(8/x) + (x(x+6))*(6)
Next, we simplify each term:
-2 * (x+6) + (-x)(x+6) = x(x(x+6)) - 8(x+6) + 6(x(x+6))
Now, distribute and combine like terms:
-2x - 12 - x^2 - 6x = x^3 + 6x^2 - 8x - 48 + 6x^2
Rearrange the terms to form a quadratic equation:
x^3 + 6x^2 - 8x - 48 + 6x^2 - x^2 - 6x -2x -12 = 0
Combine like terms:
x^3 + x^2 - 10x - 60 = 0
Now, we need to solve this cubic equation. Here are two methods you can use:
1. Graphing Calculator: Graph the equation y = x^3 + x^2 - 10x - 60 and find the x-coordinate(s) where y = 0. These x-values will give you the solutions to the equation.
2. Factoring or Rational Root Theorem: This method involves finding rational roots or factoring the cubic equation. While factoring a cubic equation can be complicated, there are online calculators available that can help you find the factors or roots.
Once we have the solutions, we can substitute them back into the original equation to ensure they are valid solutions.