If a + b + c = 0, then one of the solution of `|(a-x,c,b),(c,b-x,a),(b,a,c-x)| = 0` is

Options

A. x = a
B. `x = sqrt((3(a^(2) + b^(2) + c^(2)))/(2))`
C. `x = sqrt((2(a^(2) + b^(2) + c^(2)))/(3))`
D. x = 0

Answer

To solve the problem, we need to evaluate the determinant given the condition \(a + b + c = 0\) and find one of the solutions for the equation:

\[ \begin{vmatrix} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{vmatrix} = 0 \]

### Step 1: Apply Column Operations
We will apply the column operation \(C_1 \rightarrow C_1 + C_2 + C_3\). This means we will replace the first column with the sum of the first, second, and third columns.

After applying this operation, the first column becomes:
\[ (a - x) + c + b = a + b + c - x \]
Since \(a + b + c = 0\), this simplifies to:
\[ 0 - x = -x \]

Thus, the new first column is:
\[ \begin{pmatrix} -x \\ b - x \\ c - x \end{pmatrix} \]

### Step 2: Rewrite the Determinant
Now our determinant looks like this:
\[ \begin{vmatrix} -x & c & b \\ c & b - x & a \\ b & a & c - x \end{vmatrix} \]

### Step 3: Factor Out \(-1\)
We can factor out \(-1\) from the first column:
\[ -x \begin{vmatrix} 1 & c & b \\ c & b - x & a \\ b & a & c - x \end{vmatrix} \]

### Step 4: Set the Determinant to Zero
For the determinant to equal zero, we need:
\[ -x \cdot \text{Determinant} = 0 \]

This implies either:
1. \(x = 0\)
2. The determinant itself is zero.

### Step 5: Conclusion
Since we are looking for one of the solutions, we can conclude that:
\[ x = 0 \]
is indeed one of the solutions.

### Summary of Steps:
1. Apply the column operation to simplify the determinant.
2. Rewrite the determinant with the new first column.
3. Factor out \(-1\) and simplify.
4. Set the determinant to zero to find solutions.