If the expression `x^2+2(a+b+c)x+3(bc+ca+ab)` is a perfect square then

Options

A. a = b = c
B. `a = pm b = pm c`
C. `a = b ne c`
D. none of these

Explanation

To determine the conditions under which the expression \( x^2 + 2(a+b+c)x + 3(bc + ca + ab) \) is a perfect square, we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ f(x) = x^2 + 2(a+b+c)x + 3(bc + ca + ab) \]

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

### Step 3: Calculate the discriminant
Substituting the values into the discriminant formula:
\[ D = [2(a+b+c)]^2 - 4 \cdot 1 \cdot 3(bc + ca + ab) \]
This simplifies to:
\[ D = 4(a+b+c)^2 - 12(bc + ca + ab) \]

### Step 4: Set the discriminant to zero
For the expression to be a perfect square, we set the discriminant to zero:
\[ 4(a+b+c)^2 - 12(bc + ca + ab) = 0 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 4(a+b+c)^2 = 12(bc + ca + ab) \]
Dividing both sides by 4:
\[ (a+b+c)^2 = 3(bc + ca + ab) \]

### Step 6: Expand and simplify
Expanding the left side:
\[ a^2 + b^2 + c^2 + 2(ab + bc + ca) = 3(bc + ca + ab) \]
Rearranging gives:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]

### Step 7: Factor the expression
We can factor this expression:
\[ \frac{1}{2}(2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca) = 0 \]
This can be rewritten as:
\[ \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2] = 0 \]

### Step 8: Conclusion
The expression \( (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \) holds true if and only if:
\[ a = b = c \]

Thus, the condition for the expression \( x^2 + 2(a+b+c)x + 3(bc + ca + ab) \) to be a perfect square is that \( a, b, \) and \( c \) must be equal.

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