If a,b, c are real and `a^(2) + b ^(2) + c ^(2) = 2 (a - b - c) -3, `then the value of `a + b + c` is

Options

A. 3
B. 0
C. `-1`
D. 1

Answer

To solve the equation \( a^2 + b^2 + c^2 = 2(a - b - c) - 3 \), we will follow these steps:

### Step 1: Rearranging the equation
We start by rearranging the given equation:
\[ a^2 + b^2 + c^2 = 2a - 2b - 2c - 3 \]
We can move all terms to one side:
\[ a^2 + b^2 + c^2 - 2a + 2b + 2c + 3 = 0 \]

### Step 2: Completing the square
Next, we will complete the square for each variable \( a \), \( b \), and \( c \).

1. For \( a \):
\[ a^2 - 2a = (a - 1)^2 - 1 \]
2. For \( b \):
\[ b^2 + 2b = (b + 1)^2 - 1 \]
3. For \( c \):
\[ c^2 + 2c = (c + 1)^2 - 1 \]

Substituting these back into the equation gives:
\[ ((a - 1)^2 - 1) + ((b + 1)^2 - 1) + ((c + 1)^2 - 1) + 3 = 0 \]
This simplifies to:
\[ (a - 1)^2 + (b + 1)^2 + (c + 1)^2 = 0 \]

### Step 3: Analyzing the equation
Since the sum of squares is equal to zero, each square must be zero:
\[ (a - 1)^2 = 0 \quad \Rightarrow \quad a - 1 = 0 \quad \Rightarrow \quad a = 1 \]
\[ (b + 1)^2 = 0 \quad \Rightarrow \quad b + 1 = 0 \quad \Rightarrow \quad b = -1 \]
\[ (c + 1)^2 = 0 \quad \Rightarrow \quad c + 1 = 0 \quad \Rightarrow \quad c = -1 \]

### Step 4: Finding \( a + b + c \)
Now we can find the value of \( a + b + c \):
\[ a + b + c = 1 + (-1) + (-1) = 1 - 1 - 1 = -1 \]

### Final Answer
Thus, the value of \( a + b + c \) is:
\[ \boxed{-1} \]