If `xy = 6 and x ^(2) y + xy ^(2) + x + y = 63,` then the value of `x ^(2) + y ^(2) `is

Options

A. 61
B. 69
C. 23
D. 55

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) given that \( xy = 6 \) and \( x^2y + xy^2 + x + y = 63 \).

### Step-by-Step Solution:

1. **Start with the given equations:**
- We have \( xy = 6 \).
- We also have \( x^2y + xy^2 + x + y = 63 \).

2. **Factor the expression \( x^2y + xy^2 \):**
- Notice that \( x^2y + xy^2 \) can be factored as \( xy(x + y) \).
- Thus, we can rewrite the equation as:
\[ xy(x + y) + x + y = 63 \]
- This can be simplified to:
\[ (xy + 1)(x + y) = 63 \]

3. **Substitute the value of \( xy \):**
- Since \( xy = 6 \), we substitute this into the equation:
\[ (6 + 1)(x + y) = 63 \]
- This simplifies to:
\[ 7(x + y) = 63 \]

4. **Solve for \( x + y \):**
- Dividing both sides by 7 gives:
\[ x + y = \frac{63}{7} = 9 \]

5. **Use the identity for \( x^2 + y^2 \):**
- We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
- Substituting the values we have:
\[ x^2 + y^2 = (9)^2 - 2(6) \]

6. **Calculate \( x^2 + y^2 \):**
- This simplifies to:
\[ x^2 + y^2 = 81 - 12 = 69 \]

### Final Answer:
The value of \( x^2 + y^2 \) is \( 69 \).

---