If p and q are the root of the equation `x^(2) - px + q = 0` , then what are the vlaues of p and q respectively ?

Options

A. `1,0`
B. `0,1`
C. `-2,0`
D. `-2,1`

Explanation

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

### Step-by-Step Solution:

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

2. **Apply Vieta's formulas**:
- The sum of the roots \( p + q \) is given by:
\[ p + q = -\frac{b}{a} = -\frac{-p}{1} = p \]
- The product of the roots \( p \cdot q \) is given by:
\[ p \cdot q = \frac{c}{a} = \frac{q}{1} = q \]

3. **Set up the equations**:
- From the sum of the roots, we have:
\[ p + q = p \quad \text{(1)} \]
- From the product of the roots, we have:
\[ p \cdot q = q \quad \text{(2)} \]

4. **Analyze equation (1)**:
- Rearranging equation (1):
\[ q = p - p = 0 \]

5. **Substitute \( q = 0 \) into equation (2)**:
- Substitute \( q = 0 \) into equation (2):
\[ p \cdot 0 = 0 \quad \text{(This is always true)} \]

6. **Find the value of \( p \)**:
- Since \( q = 0 \), we can substitute back into the sum equation:
\[ p + 0 = p \quad \text{(This confirms \( p \) can be any value)} \]
- To find a specific value, we can set \( p = 1 \) (as a common choice):
\[ p = 1, \quad q = 0 \]

### Final Values:
Thus, the values of \( p \) and \( q \) are:
\[ p = 1, \quad q = 0 \]