If the expression `x^(2)-2 (Sigma a) x+3 Sigma ab=0` be a perfect square, then

Options

A. `Sigma a =0`
B. `Sigma ab=0`
C. `a=b=c`
D. none

Answer

To determine the condition under which the expression \( x^2 - 2(\Sigma a)x + 3\Sigma ab = 0 \) is a perfect square, we can follow these steps:

### Step 1: Identify the coefficients
The given expression can be rewritten in the standard quadratic form:
\[ a = 1, \quad b = -2(\Sigma a), \quad c = 3\Sigma ab \]

### Step 2: Use the condition for a perfect square
For a quadratic equation \( ax^2 + bx + c = 0 \) to be a perfect square, its discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero gives:
\[ (-2(\Sigma a))^2 - 4(1)(3\Sigma ab) = 0 \]

### Step 3: Simplify the discriminant
Calculating the discriminant:
\[ 4(\Sigma a)^2 - 12\Sigma ab = 0 \]
This simplifies to:
\[ 4(\Sigma a)^2 = 12\Sigma ab \]

### Step 4: Further simplify the equation
Dividing both sides by 4:
\[ (\Sigma a)^2 = 3\Sigma ab \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ (\Sigma a)^2 - 3\Sigma ab = 0 \]

### Step 6: Factor the equation
This can be factored as:
\[ (\Sigma a)^2 - 3\Sigma ab = (\Sigma a - \sqrt{3\Sigma ab})(\Sigma a + \sqrt{3\Sigma ab}) = 0 \]

### Step 7: Solve the factored equation
Setting each factor to zero gives us two scenarios:
1. \( \Sigma a - \sqrt{3\Sigma ab} = 0 \)
2. \( \Sigma a + \sqrt{3\Sigma ab} = 0 \) (not valid since sums of squares cannot be negative)

Thus, we focus on the first condition:
\[ \Sigma a = \sqrt{3\Sigma ab} \]

### Step 8: Square both sides
Squaring both sides leads to:
\[ (\Sigma a)^2 = 3\Sigma ab \]

### Conclusion
From the derived condition, we conclude that for the expression \( x^2 - 2(\Sigma a)x + 3\Sigma ab = 0 \) to be a perfect square, we must have:
\[ \Sigma a = \sqrt{3\Sigma ab} \]
This implies that \( a = b = c \) when all terms are equal.

### Final Answer
The condition is \( a = b = c \).

---