One root of the equation `|{:(3-x,-6,3),(-6,3-x,3),(3,3,-6-x):}|=0" "is:`

Options

A. `6`
B. `3`
C. `0`
D. None of these

Answer

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

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

### Step 2: Apply properties of determinants
We can simplify the determinant by adding the columns. Specifically, we can add the first column to the second and third columns:
\[ D = | \begin{pmatrix} 3-x & -6 + (3-x) & 3 + (3-x) \\ -6 & 3-x + (-6) & 3 + (-6) \\ 3 & 3 + (-6) & -6-x + 3 \end{pmatrix} | \]
This simplifies to:
\[ D = | \begin{pmatrix} 3-x & -3+x & 6-x \\ -6 & -3+x & -3 \\ 3 & -3 & -3-x+3 \end{pmatrix} | \]

### Step 3: Further simplify the determinant
Now we can simplify the columns:
\[ D = | \begin{pmatrix} 3-x & -3+x & 3-x \\ -6 & -3+x & 0 \\ 3 & -3 & 0 \end{pmatrix} | \]

### Step 4: Factor out common terms
We can factor out common terms from the first column:
\[ D = (3-x) | \begin{pmatrix} 1 & -3+x & 1 \\ -2 & -3+x & 0 \\ 1 & -1 & 0 \end{pmatrix} | \]

### Step 5: Calculate the determinant
Now we can calculate the determinant of the 3x3 matrix. The determinant can be calculated using the formula:
\[ D = a(ei-fh) - b(di-fg) + c(dh-eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the corresponding elements of the second and third rows.

### Step 6: Set the determinant equal to zero
We set the determinant equal to zero:
\[ (3-x) \cdot D' = 0 \]
This gives us two cases:
1. \( 3 - x = 0 \) which implies \( x = 3 \)
2. \( D' = 0 \)

### Step 7: Solve for the roots
From the first case, we have one root \( x = 3 \). Now we need to check if there are any other roots from \( D' = 0 \).

### Step 8: Check the options
The options given are:
1. 6
2. 3
3. 0
4. None of these

Since we found one root \( x = 3 \), we can conclude that one of the roots of the equation is:
\[ \boxed{3} \]