If `a-b =-7 and a ^(2) + b ^(2) = 85,` then find ab.

Options

A. A)`18`
B. B)`-30`
C. C)`-44`
D. D)`-60`

Answer

To solve the problem, we need to find the value of \( ab \) given the equations \( a - b = -7 \) and \( a^2 + b^2 = 85 \).

### Step-by-Step Solution:

1. **Start with the first equation:**
\[ a - b = -7 \]
We can rearrange this to express \( a \) in terms of \( b \):
\[ a = b - 7 \]

2. **Substitute \( a \) in the second equation:**
Now, we substitute \( a \) in the second equation \( a^2 + b^2 = 85 \):
\[ (b - 7)^2 + b^2 = 85 \]

3. **Expand the squared term:**
Expanding \( (b - 7)^2 \):
\[ b^2 - 14b + 49 + b^2 = 85 \]
Combine like terms:
\[ 2b^2 - 14b + 49 = 85 \]

4. **Rearrange the equation:**
Move 85 to the left side:
\[ 2b^2 - 14b + 49 - 85 = 0 \]
Simplifying gives:
\[ 2b^2 - 14b - 36 = 0 \]

5. **Divide the entire equation by 2:**
To simplify, divide the equation by 2:
\[ b^2 - 7b - 18 = 0 \]

6. **Factor the quadratic equation:**
We need to factor \( b^2 - 7b - 18 \):
\[ (b - 9)(b + 2) = 0 \]
So, the solutions for \( b \) are:
\[ b = 9 \quad \text{or} \quad b = -2 \]

7. **Find corresponding values of \( a \):**
For \( b = 9 \):
\[ a = 9 - 7 = 2 \]
For \( b = -2 \):
\[ a = -2 - 7 = -9 \]

8. **Calculate \( ab \):**
Now, we calculate \( ab \) for both pairs:
- For \( (a, b) = (2, 9) \):
\[ ab = 2 \times 9 = 18 \]
- For \( (a, b) = (-9, -2) \):
\[ ab = -9 \times -2 = 18 \]

Thus, in both cases, we find:
\[ ab = 18 \]

### Final Answer:
The value of \( ab \) is \( 18 \).