If `x + y = 12 and xy =32,` then what is the value of `x ^(2) + y^(2)`

Options

A. 24
B. 144
C. 128
D. 80

Answer

To find the value of \( x^2 + y^2 \) given that \( x + y = 12 \) and \( xy = 32 \), we can use the identity:

\[ x^2 + y^2 = (x + y)^2 - 2xy \]

### Step-by-Step Solution:

1. **Write down the given equations:**
\[ x + y = 12 \]
\[ xy = 32 \]

2. **Use the identity for \( x^2 + y^2 \):**
\[ x^2 + y^2 = (x + y)^2 - 2xy \]

3. **Substitute \( x + y \) into the equation:**
\[ x^2 + y^2 = (12)^2 - 2xy \]

4. **Calculate \( (12)^2 \):**
\[ (12)^2 = 144 \]

5. **Substitute \( xy \) into the equation:**
\[ x^2 + y^2 = 144 - 2(32) \]

6. **Calculate \( 2 \times 32 \):**
\[ 2 \times 32 = 64 \]

7. **Subtract \( 64 \) from \( 144 \):**
\[ x^2 + y^2 = 144 - 64 = 80 \]

8. **Final result:**
\[ x^2 + y^2 = 80 \]

### Conclusion:
The value of \( x^2 + y^2 \) is \( 80 \).