Let's solve the equation step by step:
The given equation is:
\[ \frac{1}{x-5} = \frac{1}{x+3} + \frac{x-1}{x^2-2x-15} \]
First, let's simplify \(x^2 - 2x - 15\). This can be factored:
\[ x^2 - 2x - 15 = (x - 5)(x + 3) \]
So we have:
\[ \frac{1}{x-5} = \frac{1}{x+3} + \frac{x-1}{(x-5)(x+3)} \]
Next, to combine the right-hand side, we need a common denominator, which is \((x-5)(x+3)\):
\[ \frac{1}{x-5} = \frac{(x-5) + (x-1)}{(x-5)(x+3)} \]
Simplifying the numerator on the right:
\[ (x - 5) + (x - 1) = 2x - 6 \]
Thus, the equation becomes:
\[ \frac{1}{x-5} = \frac{2x - 6}{(x-5)(x + 3)} \]
Cross-multiply:
\[ 1 \cdot (x - 5)(x + 3) = (2x - 6)(1) \]
This simplifies to:
\[ (x - 5)(x + 3) = 2x - 6 \]
Expanding the left-hand side:
\[ x^2 + 3x - 5x - 15 = 2x - 6 \]
Which simplifies to:
\[ x^2 - 2x - 15 = 2x - 6 \]
Rearranging gives us:
\[ x^2 - 2x - 2x - 15 + 6 = 0 \]
This can be simplified to:
\[ x^2 - 4x - 9 = 0 \]
Now we can use the quadratic formula to solve for \(x\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{4 \pm \sqrt{16 + 36}}{2} = \frac{4 \pm \sqrt{52}}{2} = \frac{4 \pm 2\sqrt{13}}{2} \]
Simplifying gives:
\[ x = 2 \pm \sqrt{13} \]
These solutions do not match any of the choices provided (5, -3, 1, 9, ±9).
Therefore, none of the provided options are solutions for the original equation. If there is an error in the transcription of the equation, please check it for accuracy. Otherwise, the solutions are:
\[ x = 2 + \sqrt{13} \quad \text{and} \quad x = 2 - \sqrt{13} \]
1 answer