If `a - b = 3 and a ^(2) + b ^(2) = 25,` then the value of ab is

Options

A. 16
B. 8
C. 10
D. 15

Answer

To solve the problem, we start with the given equations:

1. \( a - b = 3 \) (Equation 1)
2. \( a^2 + b^2 = 25 \) (Equation 2)

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

### Step 1: Square Equation 1
First, we square Equation 1:

\[ (a - b)^2 = 3^2 \]

This simplifies to:

\[ a^2 - 2ab + b^2 = 9 \quad \text{(Equation 3)} \]

### Step 2: Substitute Equation 2 into Equation 3
Now, we can substitute \( a^2 + b^2 \) from Equation 2 into Equation 3. Since \( a^2 + b^2 = 25 \), we can rewrite Equation 3 as:

\[ 25 - 2ab = 9 \]

### Step 3: Solve for \( ab \)
Now, we can solve for \( ab \):

\[ 25 - 2ab = 9 \]

Subtract 25 from both sides:

\[ -2ab = 9 - 25 \]

This simplifies to:

\[ -2ab = -16 \]

Now, divide both sides by -2:

\[ ab = \frac{-16}{-2} = 8 \]

### Final Answer
Thus, the value of \( ab \) is:

\[ \boxed{8} \]

---