If every pair from among the equations `x^(2)+ax+bc=0, x^(2)+bx+ca=0 and x^(2)+cx+ab=0` has a common root, then the sum and product of the three common roots is

Options

A. `2(a+b+c)`
B. a+b+c
C. `-(1)/(2)(a+b+c)`
D. abc

Answer

To solve the problem, we need to analyze the three quadratic equations given and find the common roots. The equations are:

1. \( x^2 + ax + bc = 0 \)
2. \( x^2 + bx + ca = 0 \)
3. \( x^2 + cx + ab = 0 \)

Since each pair of equations has a common root, we can denote the common roots as follows:

Let:
- \( \alpha \) be the common root of the first and second equations.
- \( \beta \) be the common root of the second and third equations.
- \( \gamma \) be the common root of the first and third equations.

### Step 1: Finding Relationships from the First Equation

For the first equation \( x^2 + ax + bc = 0 \):
- The sum of the roots \( \alpha + \beta = -a \)
- The product of the roots \( \alpha \cdot \beta = bc \)

### Step 2: Finding Relationships from the Second Equation

For the second equation \( x^2 + bx + ca = 0 \):
- The sum of the roots \( \beta + \gamma = -b \)
- The product of the roots \( \beta \cdot \gamma = ca \)

### Step 3: Finding Relationships from the Third Equation

For the third equation \( x^2 + cx + ab = 0 \):
- The sum of the roots \( \gamma + \alpha = -c \)
- The product of the roots \( \gamma \cdot \alpha = ab \)

### Step 4: Summing the Roots

Now, we can sum all three equations for the roots:

\[ (\alpha + \beta) + (\beta + \gamma) + (\gamma + \alpha) = -a - b - c \]

This simplifies to:

\[ 2(\alpha + \beta + \gamma) = -a - b - c \]

Thus, we can find the sum of the roots:

\[ \alpha + \beta + \gamma = -\frac{a + b + c}{2} \]

### Step 5: Finding the Product of the Roots

Now, we can find the product of the roots by multiplying the products from each equation:

\[ \alpha \cdot \beta \cdot \gamma = \sqrt{(bc)(ca)(ab)} \]

This can be simplified as follows:

\[ \alpha \cdot \beta \cdot \gamma = \sqrt{(abc)^2} = abc \]

### Conclusion

Thus, the sum of the three common roots is:

\[ \text{Sum} = -\frac{a + b + c}{2} \]

And the product of the three common roots is:

\[ \text{Product} = abc \]