If p,q,r are three distinct real numbers, `p ne 0` such that `x^(2)+qx +pr =0 and x^(2)+rx +pq =0` have a common root, then the value of p+q+r is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Equations**:
We have two equations:
\[ (1) \quad x^2 + qx + pr = 0 \]
\[ (2) \quad x^2 + rx + pq = 0 \]

2. **Let the Common Root be \( \alpha \)**:
Assume \( \alpha \) is the common root of both equations. Then, substituting \( \alpha \) into both equations gives us:
\[ \alpha^2 + q\alpha + pr = 0 \quad (3) \]
\[ \alpha^2 + r\alpha + pq = 0 \quad (4) \]

3. **Set the Two Equations Equal**:
Since both equations equal zero, we can set them equal to each other:
\[ q\alpha + pr = r\alpha + pq \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ q\alpha - r\alpha = pq - pr \]
\[ (q - r)\alpha = p(q - r) \]

5. **Factor Out \( (q - r) \)**:
Since \( p, q, r \) are distinct and \( p \neq 0 \), we can divide both sides by \( (q - r) \) (as long as \( q \neq r \)):
\[ \alpha = p \]

6. **Substituting \( \alpha \) Back**:
Now substitute \( \alpha = p \) back into either equation (let's use equation (3)):
\[ p^2 + qp + pr = 0 \]

7. **Rearranging**:
Rearranging gives:
\[ p^2 + qp + pr = 0 \]

8. **Using the Second Equation**:
Substitute \( \alpha = p \) into the second equation (equation (4)):
\[ p^2 + rp + pq = 0 \]

9. **Setting Up the System of Equations**:
Now we have two equations:
\[ p^2 + qp + pr = 0 \quad (5) \]
\[ p^2 + rp + pq = 0 \quad (6) \]

10. **Subtracting Equations (5) and (6)**:
Subtract equation (6) from equation (5):
\[ (q - r)p + (pr - pq) = 0 \]
\[ (q - r)p = pq - pr \]

11. **Factoring Out \( p \)**:
Rearranging gives:
\[ (q - r)p = p(q - r) \]

12. **Dividing by \( (q - r) \)**:
Since \( q \neq r \), we can divide both sides by \( (q - r) \):
\[ p = p \]
This does not provide new information.

13. **Finding \( p + q + r \)**:
From earlier, we can derive:
\[ p + q + r = 0 \]

### Conclusion:
Thus, the value of \( p + q + r \) is:
\[ \boxed{0} \]