Simplify : `1/(x^2 - 8x + 15) - 1/(x^2 - 25)`

Answer

To simplify the expression \( \frac{1}{x^2 - 8x + 15} - \frac{1}{x^2 - 25} \), we will follow these steps:

### Step 1: Factor the denominators
First, we need to factor the quadratic expressions in the denominators.

1. **For \( x^2 - 8x + 15 \)**:
- We look for two numbers that multiply to \( 15 \) (the constant term) and add up to \( -8 \) (the coefficient of \( x \)).
- The numbers are \( -3 \) and \( -5 \).
- Thus, \( x^2 - 8x + 15 = (x - 3)(x - 5) \).

2. **For \( x^2 - 25 \)**:
- This is a difference of squares, which can be factored as:
- \( x^2 - 25 = (x - 5)(x + 5) \).

### Step 2: Rewrite the expression with factored denominators
Now we can rewrite the original expression using the factored forms:

\[ \frac{1}{(x - 3)(x - 5)} - \frac{1}{(x - 5)(x + 5)} \]

### Step 3: Find a common denominator
The common denominator for the two fractions is \( (x - 3)(x - 5)(x + 5) \).

### Step 4: Rewrite each fraction with the common denominator
Now we can rewrite each fraction:

\[ \frac{(x + 5)}{(x - 3)(x - 5)(x + 5)} - \frac{(x - 3)}{(x - 5)(x + 5)(x - 3)} \]

### Step 5: Combine the fractions
Now that both fractions have a common denominator, we can combine them:

\[ \frac{(x + 5) - (x - 3)}{(x - 3)(x - 5)(x + 5)} \]

### Step 6: Simplify the numerator
Now, simplify the numerator:

\[ (x + 5) - (x - 3) = x + 5 - x + 3 = 8 \]

### Step 7: Write the simplified expression
Thus, the expression simplifies to:

\[ \frac{8}{(x - 3)(x - 5)(x + 5)} \]

### Final Answer
The simplified expression is:

\[ \frac{8}{(x - 3)(x - 5)(x + 5)} \]

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