If p and q are the roots of `x^2 + px + q = 0`, then find p.

Options

A. p = 1
B. p = 1 or 0
C. p = -2
D. p = -2 or 0

Answer

To find the value of \( p \) given that \( p \) and \( q \) are the roots of the equation \( x^2 + px + q = 0 \), we can use the relationships derived from Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.

### Step-by-Step Solution:

1. **Identify the equation and roots**:
The given quadratic equation is:
\[ x^2 + px + q = 0 \]
The roots of this equation are \( p \) and \( q \).

2. **Apply Vieta's Formulas**:
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( (p + q) \) is given by:
\[ p + q = -\frac{b}{a} \]
- The product of the roots \( (pq) \) is given by:
\[ pq = \frac{c}{a} \]
In our case, \( a = 1 \), \( b = p \), and \( c = q \). Therefore:
\[ p + q = -p \quad \text{(1)} \]
\[ pq = q \quad \text{(2)} \]

3. **From the sum of roots equation**:
From equation (1):
\[ p + q = -p \]
Rearranging gives:
\[ 2p + q = 0 \quad \Rightarrow \quad q = -2p \quad \text{(3)} \]

4. **From the product of roots equation**:
From equation (2):
\[ pq = q \]
Rearranging gives:
\[ pq - q = 0 \quad \Rightarrow \quad q(p - 1) = 0 \]
This implies either:
- \( q = 0 \) or
- \( p - 1 = 0 \) (i.e., \( p = 1 \)).

5. **Case Analysis**:
- **Case 1**: If \( q = 0 \):
Substituting \( q = 0 \) into equation (3):
\[ 0 = -2p \quad \Rightarrow \quad p = 0 \]
- **Case 2**: If \( p = 1 \):
This gives us one possible value for \( p \).

6. **Conclusion**:
The possible values of \( p \) are:
\[ p = 0 \quad \text{or} \quad p = 1 \]

### Final Answer:
The possible values of \( p \) are \( 0 \) and \( 1 \).