If `a^(2) + b^(2) + c^(2) - ab - bc - ca = 0`, then

Options

A. `a+ b + c =0`
B. `a =b = c`
C. `a = b = - c`
D. `a = -b = c`

Answer

To solve the equation \( a^2 + b^2 + c^2 - ab - bc - ca = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]

### Step 2: Multiply the equation by 2
To simplify the expression, we can multiply the entire equation by 2:
\[ 2(a^2 + b^2 + c^2 - ab - bc - ca) = 0 \]
This simplifies to:
\[ 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0 \]

### Step 3: Rearrange the terms
We can rearrange the terms as follows:
\[ (2a^2 - 2ab) + (2b^2 - 2bc) + (2c^2 - 2ca) = 0 \]

### Step 4: Factor the expression
Now, we can factor each of the terms:
\[ 2(a^2 - ab) + 2(b^2 - bc) + 2(c^2 - ca) = 0 \]
This can be rewritten as:
\[ 2[(a^2 - ab) + (b^2 - bc) + (c^2 - ca)] = 0 \]

### Step 5: Recognize the perfect square
Notice that:
\[ a^2 - ab = \frac{1}{2}(a - b)^2 \]
Thus, we can rewrite the entire expression:
\[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 0 \]

### Step 6: Set each squared term to zero
Since the sum of squares equals zero, each individual square must also be zero:
\[ (a - b)^2 = 0, \quad (b - c)^2 = 0, \quad (c - a)^2 = 0 \]

### Step 7: Conclude the equality
From these equations, we conclude:
\[ a - b = 0 \implies a = b \]
\[ b - c = 0 \implies b = c \]
\[ c - a = 0 \implies c = a \]
Thus, we find that:
\[ a = b = c \]

### Final Answer
The solution to the equation \( a^2 + b^2 + c^2 - ab - bc - ca = 0 \) is that \( a = b = c \).

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