If `a + b = 11 and ab = 15,` then `a ^(2) + b ^(2) ` is equal to :

Options

A. 90
B. 91
C. 93
D. 92

Answer

To find the value of \( a^2 + b^2 \) given that \( a + b = 11 \) and \( ab = 15 \), we can use the identity:

\[ a^2 + b^2 = (a + b)^2 - 2ab \]

### Step-by-step solution:

1. **Identify the given values**:
- \( a + b = 11 \)
- \( ab = 15 \)

2. **Calculate \( (a + b)^2 \)**:
\[ (a + b)^2 = 11^2 = 121 \]

3. **Calculate \( 2ab \)**:
\[ 2ab = 2 \times 15 = 30 \]

4. **Substitute these values into the identity**:
\[ a^2 + b^2 = (a + b)^2 - 2ab = 121 - 30 \]

5. **Perform the subtraction**:
\[ a^2 + b^2 = 121 - 30 = 91 \]

Thus, the value of \( a^2 + b^2 \) is \( 91 \).

### Final Answer:
\[ a^2 + b^2 = 91 \]