If a - b = 5 and `a^(2)+b^(2)=45` then the value of ab is:

Options

A. 25
B. 10
C. 15
D. 20

Answer

To solve the problem, we need to find the value of \( ab \) given the equations \( a - b = 5 \) and \( a^2 + b^2 = 45 \).

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ a - b = 5 \quad \text{(1)} \]
\[ a^2 + b^2 = 45 \quad \text{(2)} \]

2. **Square equation (1):**
\[ (a - b)^2 = 5^2 \]
Expanding the left side:
\[ a^2 - 2ab + b^2 = 25 \quad \text{(3)} \]

3. **Substitute \( a^2 + b^2 \) from equation (2) into equation (3):**
From equation (2), we know:
\[ a^2 + b^2 = 45 \]
Substitute this into equation (3):
\[ 45 - 2ab = 25 \]

4. **Rearranging the equation to isolate \( ab \):**
\[ -2ab = 25 - 45 \]
\[ -2ab = -20 \]
Dividing both sides by -2:
\[ ab = 10 \]

5. **Final answer:**
The value of \( ab \) is:
\[ \boxed{10} \]