If `a^(2)+b^(2)+c^(2)-2a-4b6c= -14` then `a+b+c =`

Explanation

To solve the equation \( a^2 + b^2 + c^2 - 2a - 4b - 6c = -14 \) and find the value of \( a + b + c \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation:
\[ a^2 - 2a + b^2 - 4b + c^2 - 6c + 14 = 0 \]

### Step 2: Complete the square for each variable
Now, we will complete the square for each of the variables \( a \), \( b \), and \( c \).

1. For \( a^2 - 2a \):
\[ a^2 - 2a = (a - 1)^2 - 1 \]

2. For \( b^2 - 4b \):
\[ b^2 - 4b = (b - 2)^2 - 4 \]

3. For \( c^2 - 6c \):
\[ c^2 - 6c = (c - 3)^2 - 9 \]

### Step 3: Substitute back into the equation
Now substitute these completed squares back into the equation:
\[ ((a - 1)^2 - 1) + ((b - 2)^2 - 4) + ((c - 3)^2 - 9) + 14 = 0 \]
Simplifying this gives:
\[ (a - 1)^2 + (b - 2)^2 + (c - 3)^2 - 1 - 4 - 9 + 14 = 0 \]
\[ (a - 1)^2 + (b - 2)^2 + (c - 3)^2 = 0 \]

### Step 4: Analyze the equation
Since the sum of squares equals zero, each square must individually equal zero:
\[ (a - 1)^2 = 0 \quad \Rightarrow \quad a - 1 = 0 \quad \Rightarrow \quad a = 1 \]
\[ (b - 2)^2 = 0 \quad \Rightarrow \quad b - 2 = 0 \quad \Rightarrow \quad b = 2 \]
\[ (c - 3)^2 = 0 \quad \Rightarrow \quad c - 3 = 0 \quad \Rightarrow \quad c = 3 \]

### Step 5: Calculate \( a + b + c \)
Now, we can find \( a + b + c \):
\[ a + b + c = 1 + 2 + 3 = 6 \]

### Final Answer
Thus, the value of \( a + b + c \) is:
\[ \boxed{6} \]