If a-b = 5 and ` a^(2) +b^(2) = 97 , ab = ` ?
A. 48
B. 32
C. 36
D. 72

Options

A. A
B. C
C. B
D. D

Answer

To solve the problem, we are given:

1. \( a - b = 5 \)
2. \( a^2 + b^2 = 97 \)

We need to find the value of \( ab \).

### Step 1: Use the identity for \( (a - b)^2 \)
We know that:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]

### Step 2: Substitute the known values
Substituting the values we have:
\[ 5^2 = 97 - 2ab \]

### Step 3: Calculate \( 5^2 \)
Calculating \( 5^2 \):
\[ 25 = 97 - 2ab \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate \( 2ab \):
\[ 2ab = 97 - 25 \]

### Step 5: Simplify the right side
Calculating \( 97 - 25 \):
\[ 2ab = 72 \]

### Step 6: Solve for \( ab \)
Now, divide both sides by 2:
\[ ab = \frac{72}{2} = 36 \]

Thus, the value of \( ab \) is \( 36 \).

### Final Answer
The answer is \( \boxed{36} \).