Final answer:
The excluded values in the given equation are 5 and -3, as they make the denominators zero. Solving the quadratic equation after clearing the denominators results in x=5 and x=-3, however since these are the excluded values, the original equation has no solution.
Step-by-step explanation:
To solve for x and identify excluded values, we first look at the denominators to determine values that would make the equation undefined. Here, x cannot be 5 or -3 because these would make the denominators zero, thus these are the excluded values. Next, we want to solve the equation:
\(\frac{x}{x-5} + \frac{8}{x+3} = \frac{13x-25}{x^2 -2x-15}\)
To simplify the problem, we notice that the denominator on the right side of the equation factors to \((x-5)(x+3)\), which is the least common denominator for all terms in the equation. Multiplying both sides of the equation by this least common denominator eliminates the fractions, resulting in:
\(x(x+3) + 8(x-5) = 13x - 25\)
Expanding and simplifying this equation, we get:
\(x^2 + 3x + 8x - 40 = 13x - 25\)
Combining like terms and bringing all terms to one side gives us a quadratic equation:
\(x^2 - 2x - 15 = 0\)
Factoring the quadratic equation, we find that:
\((x-5)(x+3) = 0\)
Thus, the values of x are 5 and -3; however, since those are the excluded values, we find that there is no solution to the original given equation that is valid within the domain of real numbers.
This is verified by substituting the values back into the original equation and seeing that they would result in division by zero, which is undefined.