If `a-b=-6 and a ^(2) + b ^(2) = 44,` then ab is

Options

A. `-4`
B. 8
C. `4`
D. `-8`

Answer

To solve the problem, we need to find the value of \( ab \) given the equations \( a - b = -6 \) and \( a^2 + b^2 = 44 \).

### Step-by-step Solution:

1. **Use the identity for \( a^2 + b^2 \)**:
We know that:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
We can rearrange this to find \( ab \):
\[ 2ab = a^2 + b^2 - (a - b)^2 \]

2. **Substitute the given values**:
We have \( a^2 + b^2 = 44 \) and \( a - b = -6 \). First, we calculate \( (a - b)^2 \):
\[ (a - b)^2 = (-6)^2 = 36 \]
Now substitute these values into the rearranged identity:
\[ 2ab = 44 - 36 \]

3. **Calculate \( 2ab \)**:
\[ 2ab = 44 - 36 = 8 \]

4. **Find \( ab \)**:
Now, divide both sides by 2 to find \( ab \):
\[ ab = \frac{8}{2} = 4 \]

Thus, the value of \( ab \) is \( 4 \).

### Final Answer:
\[ ab = 4 \]