The value of x satisfying `|{:(x-a,b,c),(a,x+b,c),(a,b,x+c):}|=0` is

Options

A. 0
B. a+b+c
C. `-(a+b+c)`
D. none of these

Answer

To solve the problem, we need to evaluate the determinant of the given matrix and set it equal to zero. The matrix is:

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

Let's denote this determinant as \( D \).

### Step 1: Write the determinant
We can express the determinant \( D \) as follows:

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

### Step 2: Expand the determinant
We can use the method of cofactor expansion along the first row to expand the determinant:

\[ D = (x - a) \begin{vmatrix} x + b & c \\ b & x + c \end{vmatrix} - b \begin{vmatrix} a & c \\ a & x + c \end{vmatrix} + c \begin{vmatrix} a & x + b \\ a & b \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
Now, we need to calculate each of the 2x2 determinants:

1. For \( \begin{vmatrix} x + b & c \\ b & x + c \end{vmatrix} \):

\[ = (x + b)(x + c) - bc = x^2 + (b + c)x + bc \]

2. For \( \begin{vmatrix} a & c \\ a & x + c \end{vmatrix} \):

\[ = a(x + c) - ac = ax \]

3. For \( \begin{vmatrix} a & x + b \\ a & b \end{vmatrix} \):

\[ = ab - a(x + b) = ab - ax - ab = -ax \]

### Step 4: Substitute back into the determinant
Now substituting these back into \( D \):

\[ D = (x - a)(x^2 + (b + c)x + bc) - b(ax) + c(-ax) \]

### Step 5: Simplify the expression
Expanding \( D \):

\[ D = (x - a)(x^2 + (b + c)x + bc) - abx + c(-ax) \]

This simplifies to:

\[ D = (x - a)(x^2 + (b + c)x + bc) - (ab + ac)x \]

### Step 6: Set the determinant to zero
To find the value of \( x \), we set \( D = 0 \):

\[ (x - a)(x^2 + (b + c)x + bc) - (ab + ac)x = 0 \]

### Step 7: Solve the equation
This is a polynomial equation in \( x \). We can factor or use the quadratic formula to find the roots.

### Step 8: Find the roots
The roots of the polynomial will give us the values of \( x \) that satisfy the original determinant equation.

### Final Step: Conclusion
The values of \( x \) that satisfy the equation will depend on the coefficients \( a, b, c \).