If (a - b) = 7 and ab = 9, then `(a^(2) + b^(2))` = ?

Options

A. 67
B. 31
C. 40
D. 58

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 \) given that \( a - b = 7 \) and \( ab = 9 \).

### Step-by-Step Solution:

1. **Use the identity for \( a^2 + b^2 \)**:
We can use the identity:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]

2. **Substitute the known values**:
We know that \( a - b = 7 \) and \( ab = 9 \). We can substitute these values into the identity:
\[ a^2 + b^2 = (7)^2 + 2(9) \]

3. **Calculate \( (7)^2 \)**:
\[ (7)^2 = 49 \]

4. **Calculate \( 2(9) \)**:
\[ 2(9) = 18 \]

5. **Add the results**:
Now, we add the two results together:
\[ a^2 + b^2 = 49 + 18 = 67 \]

Thus, the value of \( a^2 + b^2 \) is \( 67 \).

### Final Answer:
\[ \boxed{67} \]