If ` a^(2) + b^(2) + c^(2) = 2 (a - b - c) - 3 ` then the value of ` 4 a - 3 b + 5 c` is

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the equation \( a^2 + b^2 + c^2 = 2(a - b - c) - 3 \) and find the value of \( 4a - 3b + 5c \), we will follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ a^2 + b^2 + c^2 = 2(a - b - c) - 3 \]
Distributing the 2 on the right side:
\[ a^2 + b^2 + c^2 = 2a - 2b - 2c - 3 \]
Now, move all terms to one side of the equation:
\[ a^2 + b^2 + c^2 - 2a + 2b + 2c + 3 = 0 \]

### Step 2: Grouping the Terms
Rearranging the terms gives:
\[ a^2 - 2a + b^2 + 2b + c^2 + 2c + 3 = 0 \]

### Step 3: Completing the Square
Now, we will complete the square for each variable:

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

- For \( b^2 + 2b \):
\[ b^2 + 2b = (b + 1)^2 - 1 \]

- For \( c^2 + 2c \):
\[ c^2 + 2c = (c + 1)^2 - 1 \]

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

### Step 4: Solving for Variables
Since the sum of squares is zero, each square must be zero:
\[ (a - 1)^2 = 0 \implies a - 1 = 0 \implies a = 1 \]
\[ (b + 1)^2 = 0 \implies b + 1 = 0 \implies b = -1 \]
\[ (c + 1)^2 = 0 \implies c + 1 = 0 \implies c = -1 \]

### Step 5: Finding the Value of \( 4a - 3b + 5c \)
Now substitute \( a = 1 \), \( b = -1 \), and \( c = -1 \) into \( 4a - 3b + 5c \):
\[ 4a - 3b + 5c = 4(1) - 3(-1) + 5(-1) \]
Calculating each term:
\[ = 4 + 3 - 5 \]
\[ = 4 + 3 - 5 = 2 \]

### Final Answer
Thus, the value of \( 4a - 3b + 5c \) is \( \boxed{2} \).