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The given equation is:

\[
(4x - 8)^2 = 225
\]

### Step 1: Take the square root of both sides
To eliminate the square, take the square root on both sides of the equation:

\[
4x - 8 = \pm \sqrt{225}
\]

Since \( \sqrt{225} = 15 \), we have two cases to solve:

\[
4x - 8 = 15 \quad \text{or} \quad 4x - 8 = -15
\]

### Step 2: Solve each case

#### Case 1: \( 4x - 8 = 15 \)
\[
4x = 15 + 8 = 23
\]
\[
x = \frac{23}{4}
\]

#### Case 2: \( 4x - 8 = -15 \)
\[
4x = -15 + 8 = -7
\]
\[
x = \frac{-7}{4}
\]

### Final Answer:
The solutions to the equation are:

\[
x = \frac{23}{4}, \frac{-7}{4}
\]

Would you like more details or have any questions about the solution?  

Here are five follow-up questions related to this topic:
1. How do we determine when to use the positive and negative square roots?
2. What happens if the equation were a cubic equation instead of quadratic?
3. How can you factor quadratic equations instead of using square roots?
4. What are other ways to solve quadratic equations, like using the quadratic formula?
5. What is the importance of checking both solutions in quadratic equations?

**Tip:** Always remember to check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the quadratic equation (4x - 8)^2 = 225 step-by-step. This solution involves taking square roots and solving linear equations. Ideal for students in Grades 7-10 looking to strengthen their algebra skills.

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