https://file.aiquickdraw.com/mathbot/file/1725429774168vye04t9t.jpg

Let's solve the given equation:

\[
(x - 7)^2 - 54 = 0
\]

### Step 1: Add 54 to both sides of the equation.

\[
(x - 7)^2 - 54 + 54 = 0 + 54
\]

This simplifies to:

\[
(x - 7)^2 = 54
\]

### Step 2: Take the square root of both sides.

\[
x - 7 = \pm \sqrt{54}
\]

### Step 3: Simplify the square root.

\[
\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}
\]

So, the equation becomes:

\[
x - 7 = \pm 3\sqrt{6}
\]

### Step 4: Solve for \(x\).

\[
x = 7 \pm 3\sqrt{6}
\]

### Final Answer:
The solutions are:

\[
x = 7 + 3\sqrt{6} \quad \text{or} \quad x = 7 - 3\sqrt{6}
\]

Would you like further details or have any questions?

Here are five related questions you might find helpful:
1. What is the process for solving quadratic equations by completing the square?
2. How do you simplify expressions involving square roots?
3. What are the properties of square roots that are useful in solving equations?
4. Can the square root of a non-perfect square number be simplified?
5. How would you verify the solutions to this equation?

**Tip:** When solving equations with square roots, always consider both the positive and negative roots.

This page provides a step-by-step solution to the quadratic equation (x - 7)^2 - 54 = 0, focusing on key algebraic concepts like completing the square and solving square roots. Suitable for students in Grades 7-9.

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