Given A = (x + 2, -2) and B = (11, 6). Find x if AB = 17.

Answer

To solve the problem, we need to find the value of \( x \) such that the distance \( AB \) between the points \( A = (x + 2, -2) \) and \( B = (11, 6) \) is equal to 17. We will use the distance formula, which is given by:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates:**
- Point \( A \) has coordinates \( (x + 2, -2) \).
- Point \( B \) has coordinates \( (11, 6) \).

2. **Apply the distance formula:**
\[ AB = \sqrt{(11 - (x + 2))^2 + (6 - (-2))^2} \]

3. **Simplify the expression:**
- Calculate \( 11 - (x + 2) \):
\[ 11 - (x + 2) = 11 - x - 2 = 9 - x \]
- Calculate \( 6 - (-2) \):
\[ 6 - (-2) = 6 + 2 = 8 \]

4. **Substitute back into the distance formula:**
\[ AB = \sqrt{(9 - x)^2 + 8^2} \]
\[ AB = \sqrt{(9 - x)^2 + 64} \]

5. **Set the distance equal to 17:**
\[ \sqrt{(9 - x)^2 + 64} = 17 \]

6. **Square both sides to eliminate the square root:**
\[ (9 - x)^2 + 64 = 17^2 \]
\[ (9 - x)^2 + 64 = 289 \]

7. **Isolate the squared term:**
\[ (9 - x)^2 = 289 - 64 \]
\[ (9 - x)^2 = 225 \]

8. **Take the square root of both sides:**
\[ 9 - x = \pm 15 \]

9. **Solve for \( x \):**
- Case 1: \( 9 - x = 15 \)
\[ -x = 15 - 9 \]
\[ -x = 6 \quad \Rightarrow \quad x = -6 \]
- Case 2: \( 9 - x = -15 \)
\[ -x = -15 - 9 \]
\[ -x = -24 \quad \Rightarrow \quad x = 24 \]

10. **Final values of \( x \):**
\[ x = -6 \quad \text{or} \quad x = 24 \]

### Final Answer:
The values of \( x \) are \( 24 \) and \( -6 \).