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 find a root of the equation given by the determinant \(|[3-x,-6,3],[-6,3-x,3],[3,3,-6-x]|=0\), we will calculate the determinant step by step.

### Step 1: Write down 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: Expand the determinant
We can expand the determinant using the first row:
\[ D = (3-x) \begin{vmatrix} 3-x & 3 \\ 3 & -6-x \end{vmatrix} - (-6) \begin{vmatrix} -6 & 3 \\ 3 & -6-x \end{vmatrix} + 3 \begin{vmatrix} -6 & 3-x \\ 3 & 3 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
1. **First 2x2 determinant**:
\[ \begin{vmatrix} 3-x & 3 \\ 3 & -6-x \end{vmatrix} = (3-x)(-6-x) - (3)(3) = -18 - 3x + 6x + x^2 - 9 = x^2 + 3x - 27 \]

2. **Second 2x2 determinant**:
\[ \begin{vmatrix} -6 & 3 \\ 3 & -6-x \end{vmatrix} = (-6)(-6-x) - (3)(3) = 36 + 6x - 9 = 27 + 6x \]

3. **Third 2x2 determinant**:
\[ \begin{vmatrix} -6 & 3-x \\ 3 & 3 \end{vmatrix} = (-6)(3) - (3-x)(3) = -18 - (9 - 3x) = -9 + 3x \]

### Step 4: Substitute back into the determinant
Now substituting these back into the determinant:
\[ D = (3-x)(x^2 + 3x - 27) + 6(27 + 6x) + 3(-9 + 3x) \]

### Step 5: Expand and simplify
Expanding each term:
1. \( (3-x)(x^2 + 3x - 27) = 3x^2 + 9x - 81 - x^3 - 3x^2 + 27x = -x^3 + 36x - 81 \)
2. \( 6(27 + 6x) = 162 + 36x \)
3. \( 3(-9 + 3x) = -27 + 9x \)

Combining all these:
\[ D = -x^3 + 36x - 81 + 162 + 36x - 27 + 9x \]
\[ D = -x^3 + 81x + 54 \]

### Step 6: Set the determinant to zero
We need to find the roots of:
\[ -x^3 + 81x + 54 = 0 \]
or equivalently,
\[ x^3 - 81x - 54 = 0 \]

### Step 7: Factor the equation
We can use the Rational Root Theorem or trial and error to find potential roots. Testing \(x = 0\), \(x = 9\), and \(x = -9\):
1. \(x = 0\): \(0^3 - 81(0) - 54 \neq 0\)
2. \(x = 9\): \(9^3 - 81(9) - 54 = 729 - 729 - 54 = -54 \neq 0\)
3. \(x = -9\): \((-9)^3 - 81(-9) - 54 = -729 + 729 - 54 = -54 \neq 0\)

### Step 8: Identify the roots
After testing, we find that:
- The roots are \(x = 0\), \(x = 9\), and \(x = -9\).

### Final Answer
The roots of the equation are \(x = 0\), \(x = 9\), and \(x = -9\).