The problem asks to simplify two expressions. 1. Simplify $\frac{x^2-10x+25}{x^2-25}$

1. Problem Description

The problem asks to simplify two expressions.

1. Simplify $\frac{x^2-10x+25}{x^2-25}$

2. Simplify $\frac{x^2-3x}{x^2-x-2} \times \frac{x-2}{x^2-6x+9}$

2. Solution Steps

Problem 1: $\frac{x^2-10x+25}{x^2-25}$

First, factor the numerator and the denominator.

The numerator is a perfect square trinomial: $x^2 - 10x + 25 = (x-5)^2 = (x-5)(x-5)$

The denominator is a difference of squares: $x^2 - 25 = (x-5)(x+5)$

So, we have: $\frac{(x-5)(x-5)}{(x-5)(x+5)}$

Cancel the common factor $(x-5)$ from the numerator and the denominator.

$\frac{x-5}{x+5}$

Problem 2: $\frac{x^2-3x}{x^2-x-2} \times \frac{x-2}{x^2-6x+9}$

First, factor each polynomial.

$x^2 - 3x = x(x-3)$

$x^2 - x - 2 = (x-2)(x+1)$

$x-2$ remains as $x-2$

$x^2 - 6x + 9 = (x-3)^2 = (x-3)(x-3)$

So, we have: $\frac{x(x-3)}{(x-2)(x+1)} \times \frac{x-2}{(x-3)(x-3)}$

$\frac{x(x-3)(x-2)}{(x-2)(x+1)(x-3)(x-3)}$

Cancel the common factors $(x-3)$ and $(x-2)$ from the numerator and the denominator.

$\frac{x}{(x+1)(x-3)} = \frac{x}{x^2-3x+x-3} = \frac{x}{x^2-2x-3}$

3. Final Answer

Problem 1: $\frac{x-5}{x+5}$

Problem 2: $\frac{x}{(x+1)(x-3)}$ or $\frac{x}{x^2-2x-3}$