If `a^(2) + b^(2) + c^(2) + 96 = 8(a + b - 2c)`, then `sqrt(ab - bc + ca)` is equal to :

Options

A. 6
B. `2sqrt(3)`
C. `2sqrt(2)`
D. 4

Answer

To solve the equation \( a^2 + b^2 + c^2 + 96 = 8(a + b - 2c) \) and find the value of \( \sqrt{ab - bc + ca} \), we will follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation:
\[ a^2 + b^2 + c^2 + 96 - 8a - 8b + 16c = 0 \]

### Step 2: Group terms
Rearranging gives us:
\[ a^2 - 8a + b^2 - 8b + c^2 + 16c + 96 = 0 \]

### Step 3: Complete the square for \( a \)
To complete the square for \( a \):
\[ a^2 - 8a = (a - 4)^2 - 16 \]
So, substituting back:
\[ (a - 4)^2 - 16 \]

### Step 4: Complete the square for \( b \)
Now for \( b \):
\[ b^2 - 8b = (b - 4)^2 - 16 \]
Substituting back gives:
\[ (b - 4)^2 - 16 \]

### Step 5: Complete the square for \( c \)
Now for \( c \):
\[ c^2 + 16c = (c + 8)^2 - 64 \]
Substituting back gives:
\[ (c + 8)^2 - 64 \]

### Step 6: Substitute completed squares back into the equation
Now, substituting all completed squares back into the equation:
\[ ((a - 4)^2 - 16) + ((b - 4)^2 - 16) + ((c + 8)^2 - 64) + 96 = 0 \]
This simplifies to:
\[ (a - 4)^2 + (b - 4)^2 + (c + 8)^2 + 0 = 0 \]

### Step 7: Set each squared term to zero
Since the sum of squares is zero, each term must be zero:
\[ (a - 4)^2 = 0 \implies a = 4 \]
\[ (b - 4)^2 = 0 \implies b = 4 \]
\[ (c + 8)^2 = 0 \implies c = -8 \]

### Step 8: Calculate \( \sqrt{ab - bc + ca} \)
Now we substitute \( a = 4 \), \( b = 4 \), and \( c = -8 \) into \( \sqrt{ab - bc + ca} \):
\[ ab = 4 \cdot 4 = 16 \]
\[ bc = 4 \cdot (-8) = -32 \]
\[ ca = (-8) \cdot 4 = -32 \]
Now substituting these values:
\[ \sqrt{16 - (-32) + (-32)} = \sqrt{16 + 32 - 32} = \sqrt{16} = 4 \]

### Final Answer
Thus, the value of \( \sqrt{ab - bc + ca} \) is \( \boxed{4} \).