what is/are the solution to the following equation? (x-4)^2-25=0

The given equation is:

\[
(x - 4)^2 - 25 = 0
\]

### Step 1: Simplify the equation
First, add 25 to both sides:

\[
(x - 4)^2 = 25
\]

### Step 2: Solve for \( x - 4 \)
Now, take the square root of both sides:

\[
x - 4 = \pm \sqrt{25}
\]

Since \(\sqrt{25} = 5\), we get two possible cases:

\[
x - 4 = 5 \quad \text{or} \quad x - 4 = -5
\]

### Step 3: Solve for \( x \)
1. If \(x - 4 = 5\), then:
   \[
   x = 5 + 4 = 9
   \]
   
2. If \(x - 4 = -5\), then:
   \[
   x = -5 + 4 = -1
   \]

### Final Answer:
The solutions are:

\[
x = 9 \quad \text{or} \quad x = -1
\]

Would you like more details on any specific step?

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Here are 5 related questions:
1. How would you solve \((x - 3)^2 - 16 = 0\)?
2. What happens if the equation was \((x - 4)^2 + 25 = 0\)?
3. Can you solve \( (x + 1)^2 = 36 \)?
4. How does this method differ from solving a linear equation?
5. What are the key steps in solving quadratic equations by factoring?

**Tip:** Always check both positive and negative square root values when solving quadratic equations like this!

Explore how to solve the quadratic equation (x-4)^2 - 25 = 0 step-by-step. This problem covers essential algebraic concepts such as factoring, square roots, and the Zero Product Property. Ideal for students in Grades 7-9.

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