If every pair from among the equations `x^2 + px + qr = 0, x^2 + qx +rp = 0` and `x^2+rx +pq = 0` has a common root then the product of three common root is

Options

A. pqr
B. 2 pqr
C. `p^(2) q^(2) r^(2)`
D. none of these

Explanation

To solve the problem, we need to find the product of the common roots of the three given quadratic equations. Let's denote the equations as follows:

1. \( x^2 + px + qr = 0 \) (Equation 1)
2. \( x^2 + qx + rp = 0 \) (Equation 2)
3. \( x^2 + rx + pq = 0 \) (Equation 3)

### Step 1: Identify the common roots
Let’s assume that the three equations have common roots. Let’s denote the common roots as \( \alpha, \beta, \) and \( \gamma \).

### Step 2: Set up the relationships between the roots
From the equations, we know that:
- For Equation 1, if \( \alpha \) is a root, then:
\[ \alpha^2 + p\alpha + qr = 0 \quad \text{(1)} \]
- For Equation 2, if \( \beta \) is a root, then:
\[ \beta^2 + q\beta + rp = 0 \quad \text{(2)} \]
- For Equation 3, if \( \gamma \) is a root, then:
\[ \gamma^2 + r\gamma + pq = 0 \quad \text{(3)} \]

### Step 3: Use the relationships of roots
Since each pair of equations has a common root, we can assume:
- \( \beta \) is common between Equation 1 and Equation 2.
- \( \gamma \) is common between Equation 2 and Equation 3.
- \( \alpha \) is common between Equation 1 and Equation 3.

### Step 4: Express the product of the roots
Using Vieta's formulas, we can express the product of the roots for each equation:
- For Equation 1:
\[ \alpha \cdot \beta = -\frac{qr}{1} \quad \text{(4)} \]
- For Equation 2:
\[ \beta \cdot \gamma = -\frac{rp}{1} \quad \text{(5)} \]
- For Equation 3:
\[ \gamma \cdot \alpha = -\frac{pq}{1} \quad \text{(6)} \]

### Step 5: Multiply the equations
Now, we multiply the products from equations (4), (5), and (6):
\[ (\alpha \beta)(\beta \gamma)(\gamma \alpha) = (-qr)(-rp)(-pq) \]
This simplifies to:
\[ \alpha^2 \beta^2 \gamma^2 = pqr \cdot qr \cdot rp \cdot pq \]

### Step 6: Simplify the expression
This can be rewritten as:
\[ \alpha^2 \beta^2 \gamma^2 = p^2 q^2 r^2 \]

### Step 7: Take the square root
Taking the square root of both sides gives us:
\[ \alpha \beta \gamma = pqr \]

### Final Answer
Thus, the product of the three common roots is:
\[ \alpha \beta \gamma = pqr \]