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 start with the given equations:

1. \( xy = 6 \)
2. \( x^2y + xy^2 + x + y = 63 \)

We need to find the value of \( x^2 + y^2 \).

### Step 1: Rewrite the second equation
The expression \( x^2y + xy^2 \) can be factored as follows:
\[ x^2y + xy^2 = xy(x + y) \]
Thus, we can rewrite the second equation:
\[ xy(x + y) + x + y = 63 \]

### Step 2: Factor out \( x + y \)
Now, we can factor out \( x + y \) from the left side:
\[ (xy + 1)(x + y) = 63 \]

### Step 3: Substitute \( xy \)
Since we know that \( xy = 6 \), we substitute this into the equation:
\[ (6 + 1)(x + y) = 63 \]
This simplifies to:
\[ 7(x + y) = 63 \]

### Step 4: Solve for \( x + y \)
Now, we can solve for \( x + y \):
\[ x + y = \frac{63}{7} = 9 \]

### Step 5: Use the identity for \( x^2 + y^2 \)
We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the values we found:
\[ x^2 + y^2 = (9)^2 - 2(6) \]

### Step 6: Calculate \( x^2 + y^2 \)
Calculating the right side:
\[ x^2 + y^2 = 81 - 12 = 69 \]

Thus, the value of \( x^2 + y^2 \) is \( 69 \).

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

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