If `p and q(ne0)` are the roots of the equation `x^(2)+px+q=0`, then the value of p must be equal to

Options

A. a. 0, 1
B. c. `(1)/(4)`
C. b. 0
D. d. 2, 0

Answer

To solve the problem, we need to find the value of \( p \) given that \( p \) and \( q \) (where \( q \neq 0 \)) are the roots of the quadratic equation \( x^2 + px + q = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \( x^2 + px + q = 0 \). Here, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = p \) (coefficient of \( x \))
- \( c = q \) (constant term)

2. **Sum of the roots**:
The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ \text{Sum of roots} = -\frac{b}{a} \]
Substituting the values, we have:
\[ p + q = -\frac{p}{1} = -p \]
Rearranging this gives:
\[ p + q = -p \implies q = -2p \quad \text{(Equation 1)} \]

3. **Substituting one root into the equation**:
Since \( p \) is a root of the equation \( x^2 + px + q = 0 \), we substitute \( x = p \):
\[ p^2 + p \cdot p + q = 0 \]
This simplifies to:
\[ p^2 + p^2 + q = 0 \implies 2p^2 + q = 0 \]

4. **Substituting \( q \) from Equation 1**:
Now, we substitute \( q = -2p \) from Equation 1 into the equation \( 2p^2 + q = 0 \):
\[ 2p^2 - 2p = 0 \]

5. **Factoring the equation**:
We can factor out \( 2p \):
\[ 2p(p - 1) = 0 \]
This gives us two possible solutions:
\[ 2p = 0 \quad \text{or} \quad p - 1 = 0 \]

6. **Finding the values of \( p \)**:
From \( 2p = 0 \):
\[ p = 0 \]
From \( p - 1 = 0 \):
\[ p = 1 \]

7. **Conclusion**:
Thus, the possible values of \( p \) are:
\[ p = 0 \quad \text{or} \quad p = 1 \]

### Final Answer:
The values of \( p \) must be equal to \( 0 \) and \( 1 \).

---