If p and q are roots of the quadratic equation `x^(2) + mx + m^(2) + a = 0`, then the value of `p^(2) + q^(2) + pq`, is

Options

A. 0
B. a
C. `-a`
D. `+- m^(2)`

Answer

To find the value of \( p^2 + q^2 + pq \) given that \( p \) and \( q \) are roots of the quadratic equation \( x^2 + mx + m^2 + a = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients from the quadratic equation:**
The given equation is:
\[ x^2 + mx + (m^2 + a) = 0 \]
Here, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = m \) (coefficient of \( x \))
- \( c = m^2 + a \) (constant term)

2. **Use Vieta's formulas:**
According to Vieta's formulas:
- The sum of the roots \( p + q = -\frac{b}{a} = -\frac{m}{1} = -m \)
- The product of the roots \( pq = \frac{c}{a} = \frac{m^2 + a}{1} = m^2 + a \)

3. **Express \( p^2 + q^2 + pq \):**
We know that:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Therefore, we can express \( p^2 + q^2 + pq \) as:
\[ p^2 + q^2 + pq = (p + q)^2 - 2pq + pq = (p + q)^2 - pq \]

4. **Substitute the values of \( p + q \) and \( pq \):**
Substitute \( p + q = -m \) and \( pq = m^2 + a \):
\[ p^2 + q^2 + pq = (-m)^2 - (m^2 + a) \]

5. **Simplify the expression:**
Now, simplify the expression:
\[ p^2 + q^2 + pq = m^2 - (m^2 + a) = m^2 - m^2 - a = -a \]

### Final Answer:
Thus, the value of \( p^2 + q^2 + pq \) is:
\[ \boxed{-a} \]