To solve the equation \( x^2 + 6x = 7 \) by completing the square, we follow these steps:

### Step-by-Step Solution

1. **Rewrite the equation:**
   \[
   x^2 + 6x = 7
   \]

2. **Move the constant term to the other side:**
   The constant is already isolated, so this step is not needed here.

3. **Complete the square on the left side:**
   - Take half of the coefficient of \(x\), which is \(6\), and divide it by 2:
     \[
     \frac{6}{2} = 3
     \]
   - Square the result:
     \[
     3^2 = 9
     \]

4. **Add \(9\) to both sides of the equation to keep it balanced:**
   \[
   x^2 + 6x + 9 = 7 + 9
   \]

5. **Simplify the equation:**
   \[
   (x + 3)^2 = 16
   \]

6. **Take the square root of both sides:**
   \[
   x + 3 = \pm\sqrt{16}
   \]
   \[
   x + 3 = \pm 4
   \]

7. **Solve for \(x\):**
   - For the positive root:
     \[
     x + 3 = 4 \implies x = 4 - 3 = 1
     \]
   - For the negative root:
     \[
     x + 3 = -4 \implies x = -4 - 3 = -7
     \]

### Final Solution:
\[
x = 1 \quad \text{or} \quad x = -7
\]

Would you like further details or have any questions?

### Expanding Further:
1. How is completing the square used in solving quadratic equations?
2. What is the significance of balancing both sides of the equation?
3. How can this method be applied to equations with different coefficients?
4. What are the benefits of using the square root property in equations?
5. How do these solutions relate to graphing the function?

**Tip:** Always check your solutions by plugging them back into the original equation to verify their correctness!

Learn how to solve the quadratic equation x^2 + 6x = 7 using the completing the square method with a step-by-step solution. This problem covers key algebraic concepts, including completing the square and solving quadratic equations. Suitable for students in Grades 9-12.

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