If `p-q=6` and `p^2+q^2=116`, what is the value of pq?

Options

A. 30
B. 40
C. 20
D. 50

Answer

To solve the problem, we need to find the value of \( pq \) given the equations \( p - q = 6 \) and \( p^2 + q^2 = 116 \).

### Step-by-Step Solution:

1. **Use the identity for \( (p - q)^2 \)**:
\[ (p - q)^2 = p^2 + q^2 - 2pq \]
We know \( p - q = 6 \), so we can square both sides:
\[ (6)^2 = p^2 + q^2 - 2pq \]
This simplifies to:
\[ 36 = p^2 + q^2 - 2pq \]

2. **Substitute \( p^2 + q^2 \)**:
We are given that \( p^2 + q^2 = 116 \). Substitute this value into the equation:
\[ 36 = 116 - 2pq \]

3. **Rearranging the equation**:
To isolate \( 2pq \), we rearrange the equation:
\[ 2pq = 116 - 36 \]
Simplifying the right side gives:
\[ 2pq = 80 \]

4. **Solving for \( pq \)**:
Now, divide both sides by 2 to find \( pq \):
\[ pq = \frac{80}{2} = 40 \]

### Final Answer:
Thus, the value of \( pq \) is \( 40 \).

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