If `x+y=13 and xy = 40,` then the value of `(x^(2)+y^(2))` is :

Options

A. 69
B. 80
C. 89
D. none of these

Answer

To solve the problem where \( x + y = 13 \) and \( xy = 40 \), we need to find the value of \( x^2 + y^2 \).

### Step-by-Step Solution:

1. **Use the identity for \( x^2 + y^2 \)**:
We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
This identity will help us express \( x^2 + y^2 \) in terms of \( x + y \) and \( xy \).

2. **Substitute the known values**:
From the problem, we have:
- \( x + y = 13 \)
- \( xy = 40 \)

Now, substitute these values into the identity:
\[ x^2 + y^2 = (13)^2 - 2(40) \]

3. **Calculate \( (13)^2 \)**:
\[ (13)^2 = 169 \]

4. **Calculate \( 2(40) \)**:
\[ 2(40) = 80 \]

5. **Combine the results**:
Now, substitute back into the equation:
\[ x^2 + y^2 = 169 - 80 \]

6. **Perform the subtraction**:
\[ x^2 + y^2 = 89 \]

Thus, the value of \( x^2 + y^2 \) is \( 89 \).

### Final Answer:
\[ \boxed{89} \]