If `a +b =- 8 and a ^(2) + b ^(2) = 43,` then value of ab is

Options

A. 30
B. 64
C. 98
D. 10.5

Answer

To find the value of \( ab \) given the equations \( a + b = -8 \) and \( a^2 + b^2 = 43 \), we can follow these steps:

### Step 1: Use the identity for the square of a sum
We know that:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
Substituting the known values:
\[ (-8)^2 = a^2 + 2ab + b^2 \]
Calculating \( (-8)^2 \):
\[ 64 = a^2 + 2ab + b^2 \]

### Step 2: Substitute \( a^2 + b^2 \)
We have \( a^2 + b^2 = 43 \). Substitute this into the equation from Step 1:
\[ 64 = 43 + 2ab \]

### Step 3: Solve for \( 2ab \)
Rearranging the equation gives:
\[ 2ab = 64 - 43 \]
Calculating the right side:
\[ 2ab = 21 \]

### Step 4: Solve for \( ab \)
Now, divide both sides by 2:
\[ ab = \frac{21}{2} = 10.5 \]

Thus, the value of \( ab \) is \( 10.5 \).