A root of the equation `|(3-x,-6,3),(-6,3-x,3),(3,3,-6-x)|` = 0

Options

A. 6
B. 3
C. 0
D. none of these

Answer

To solve the equation given by the determinant \( |(3-x,-6,3),(-6,3-x,3),(3,3,-6-x)| = 0 \), we will follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} 3-x & -6 & 3 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} \]

### Step 2: Apply Row Operations
We can simplify the determinant by applying row operations. Let's add the second and third rows to the first row:
\[ R_1 \rightarrow R_1 + R_2 + R_3 \]
Calculating the new first row:
- First element: \( (3-x) + (-6) + 3 = (3-x - 6 + 3) = -x \)
- Second element: \( -6 + (3-x) + 3 = (-6 + 3 - x + 3) = -x \)
- Third element: \( 3 + 3 + (-6-x) = (3 + 3 - 6 - x) = -x \)

Thus, the determinant becomes:
\[ D = \begin{vmatrix} -x & -x & -x \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} \]

### Step 3: Factor Out \(-x\)
We can factor out \(-x\) from the first row:
\[ D = -x \begin{vmatrix} 1 & 1 & 1 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} \]

### Step 4: Set the Determinant Equal to Zero
Now, we set the determinant equal to zero:
\[ -x \cdot \begin{vmatrix} 1 & 1 & 1 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} = 0 \]

### Step 5: Solve for Roots
The product of two factors is zero, which means either:
1. \(-x = 0\) or
2. \(\begin{vmatrix} 1 & 1 & 1 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} = 0\)

From the first factor, we find:
\[ x = 0 \]

### Step 6: Check the Second Factor
We can check if the second determinant can yield any roots, but since we only need one root, we can conclude that \(x = 0\) is a valid root.

### Conclusion
Thus, one root of the given determinant equation is:
\[ \boxed{0} \]