If the roots of the equation `(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0` are equal then

Options

A. a + b + c = 0
B. `a + b omega + c omega^(2) = 0`
C. a - b + c = 0
D. none of these

Explanation

To solve the equation \((x-b)(x-c) + (x-c)(x-a) + (x-a)(x-b) = 0\) and determine the condition under which the roots are equal, we can follow these steps:

### Step 1: Expand the given equation
We start by expanding each term in the equation:

1. \((x-b)(x-c) = x^2 - (b+c)x + bc\)
2. \((x-c)(x-a) = x^2 - (c+a)x + ca\)
3. \((x-a)(x-b) = x^2 - (a+b)x + ab\)

Now, we can add these three expressions together:

\[ (x-b)(x-c) + (x-c)(x-a) + (x-a)(x-b) = (x^2 - (b+c)x + bc) + (x^2 - (c+a)x + ca) + (x^2 - (a+b)x + ab) \]

Combining like terms, we have:

\[ 3x^2 - [(b+c) + (c+a) + (a+b)]x + (bc + ca + ab) = 0 \]

This simplifies to:

\[ 3x^2 - 2(a+b+c)x + (ab + ac + bc) = 0 \]

### Step 2: Set the discriminant to zero
For the roots of a quadratic equation \(Ax^2 + Bx + C = 0\) to be equal, the discriminant must be zero. The discriminant \(D\) is given by:

\[ D = B^2 - 4AC \]

In our case, \(A = 3\), \(B = -2(a+b+c)\), and \(C = ab + ac + bc\). Thus, we have:

\[ D = [-2(a+b+c)]^2 - 4 \cdot 3 \cdot (ab + ac + bc) \]

Calculating this gives:

\[ D = 4(a+b+c)^2 - 12(ab + ac + bc) \]

Setting the discriminant to zero, we get:

\[ 4(a+b+c)^2 - 12(ab + ac + bc) = 0 \]

### Step 3: Rearranging the equation
Rearranging the equation yields:

\[ 4(a+b+c)^2 = 12(ab + ac + bc) \]

Dividing both sides by 4 gives:

\[ (a+b+c)^2 = 3(ab + ac + bc) \]

### Step 4: Use the identity
Using the identity \((a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc)\), we can substitute:

\[ a^2 + b^2 + c^2 + 2(ab + ac + bc) = 3(ab + ac + bc) \]

Rearranging this results in:

\[ a^2 + b^2 + c^2 - ab - ac - bc = 0 \]

### Step 5: Factor the equation
This can be factored as:

\[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \]

Since squares of real numbers are non-negative, the only solution is when each square is zero:

\[ a-b = 0, \quad b-c = 0, \quad c-a = 0 \]

This implies:

\[ a = b = c \]

### Conclusion
Thus, the condition for the roots of the given equation to be equal is:

\[ a + b + c = 0 \quad \text{(Option A)} \]