If the equation `x^(2)+px+q=0 and x^(2)+qx+p=0` have a common root then 1+p+q =

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \(1 + p + q\) given that the equations \(x^2 + px + q = 0\) and \(x^2 + qx + p = 0\) have a common root.

### Step-by-Step Solution:

1. **Assume a Common Root**:
Let \(\alpha\) be the common root of both equations. Therefore, \(\alpha\) satisfies both equations:
\[ \alpha^2 + p\alpha + q = 0 \quad \text{(1)} \]
\[ \alpha^2 + q\alpha + p = 0 \quad \text{(2)} \]

2. **Equate the Two Equations**:
Since both equations equal zero, we can set them equal to each other:
\[ p\alpha + q = q\alpha + p \]

3. **Rearranging the Equation**:
Rearranging gives us:
\[ p\alpha - q\alpha = p - q \]
\[ (p - q)\alpha = p - q \]

4. **Case Analysis**:
- If \(p \neq q\), we can divide both sides by \(p - q\):
\[ \alpha = 1 \]
- If \(p = q\), then both equations become the same, and we can proceed with \(p\) and \(q\) being equal.

5. **Substituting \(\alpha = 1\)**:
Substitute \(\alpha = 1\) back into either equation (let's use equation (1)):
\[ 1^2 + p(1) + q = 0 \]
\[ 1 + p + q = 0 \]

6. **Finding \(1 + p + q\)**:
Rearranging gives us:
\[ 1 + p + q = 0 \]
Thus, we find:
\[ 1 + p + q = 0 \]

### Conclusion:
The value of \(1 + p + q\) is:
\[ \boxed{0} \]