Which of the following tables shows the correct steps to transform x2 + 6x + 8 = 0 into the form (x − p)2 = q?[p and q are integers]

**Step 1**

x2 + 6x + 8 − 1 = 0 − 1

**Step 2**

x2 + 6x + 7 = −1

**Step 3**

(x + 3)2 = −1

**Step 1**

x2 + 6x + 8 + 1 = 0 + 1

**Step 2**

x2 + 6x + 9 = 1

**Step 3**

(x + 3)2 = 1

**Step 1**

x2 + 6x + 8 − 2 = 0 − 2

**Step 2**

x2 + 6x + 6 = −2

**Step 3**

(x + 3)2 = −2

**Step 1**

x2 + 6x + 8 + 2 = 0 + 2

**Step 2**

x2 + 6x + 10 = 2

**Step 3**

(x + 3)2 = 2


The goal is to complete the square for the quadratic equation \( x^2 + 6x + 8 = 0 \) and transform it into the form \( (x - p)^2 = q \).

### Step-by-Step Process:

1. **Start with the original equation:**
   \[
   x^2 + 6x + 8 = 0
   \]

2. **Move the constant term (8) to the other side of the equation:**
   \[
   x^2 + 6x = -8
   \]

3. **Complete the square:**
   - Take the coefficient of \( x \), which is 6, divide it by 2, and square it:
     \[
     \left(\frac{6}{2}\right)^2 = 9
     \]
   - Add 9 to both sides of the equation to complete the square:
     \[
     x^2 + 6x + 9 = -8 + 9
     \]

4. **Simplify the equation:**
   \[
   (x + 3)^2 = 1
   \]

### Analyzing the Given Tables:

- **Table 2 (Second Option):**
  - **Step 1:** \( x^2 + 6x + 8 + 1 = 0 + 1 \)
  - **Step 2:** \( x^2 + 6x + 9 = 1 \)
  - **Step 3:** \( (x + 3)^2 = 1 \)

This sequence correctly follows the steps to complete the square and transform the equation into the form \( (x - p)^2 = q \), where \( p = -3 \) and \( q = 1 \).

### Conclusion:
The correct table that shows the steps to transform \( x^2 + 6x + 8 = 0 \) into the form \( (x - p)^2 = q \) is **Table 2**.

Would you like to dive deeper into the method of completing the square, or do you have any questions?

Here are 8 related questions to expand your understanding:
1. What is the geometric interpretation of completing the square?
2. How can you use completing the square to derive the quadratic formula?
3. What would the equation look like if \( x^2 + 6x + 8 = 0 \) were a maximization problem?
4. How would the process change if the coefficient of \( x^2 \) were different from 1?
5. What are some real-world applications of solving quadratic equations by completing the square?
6. How does completing the square help in solving quadratic equations graphically?
7. What is the effect of different values of \( q \) in the equation \( (x - p)^2 = q \)?
8. Can you derive the vertex form of a quadratic equation using completing the square?

**Tip:** Completing the square is not only useful for solving quadratic equations but also for understanding the geometry of parabolas in algebraic contexts.

Learn how to transform the quadratic equation x^2 + 6x + 8 = 0 into the vertex form (x - p)^2 = q step-by-step. Understand the concept of completing the square, find values of p and q, and solve quadratic equations effectively. Suitable for advanced high school students.

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