## Factoring and Solving (x-4)^2 - 25

The expression (x-4)^2 - 25 represents a quadratic equation in a slightly disguised form. Let's break down how to factor and solve it:

### Recognizing the Difference of Squares

The key to simplifying this expression lies in recognizing the pattern of **difference of squares**:

**a^2 - b^2 = (a + b)(a - b)**

In our case, we can rewrite (x-4)^2 as (x-4)(x-4). Therefore:

(x-4)^2 - 25 = (x-4)(x-4) - 5^2

Now we can clearly see the difference of squares pattern with a = (x-4) and b = 5.

### Factoring the Expression

Applying the difference of squares formula:

(x-4)(x-4) - 5^2 = **[(x-4) + 5][(x-4) - 5]**

Simplifying the expressions inside the brackets:

**= (x + 1)(x - 9)**

### Solving for x

To find the values of x that satisfy the equation (x-4)^2 - 25 = 0, we set the factored expression equal to zero:

(x + 1)(x - 9) = 0

This equation holds true when either factor is equal to zero:

**x + 1 = 0 => x = -1****x - 9 = 0 => x = 9**

Therefore, the solutions to the equation (x-4)^2 - 25 = 0 are **x = -1** and **x = 9**.

### Summary

We successfully factored and solved the expression (x-4)^2 - 25 by recognizing the difference of squares pattern. This method allows us to simplify the expression and find the solutions to the equation.