If `|(1,2,3),(2,x,3),(3,4,5)|=0` then x=________

Answer

To solve the determinant equation \(|(1,2,3),(2,x,3),(3,4,5)|=0\), we will follow these steps:

### Step 1: Write the determinant
We start with the determinant of the given matrix:

\[ D = \begin{vmatrix} 1 & 2 & 3 \\ 2 & x & 3 \\ 3 & 4 & 5 \end{vmatrix} \]

### Step 2: Expand the determinant using the first row
We will expand the determinant along the first row:

\[ D = 1 \cdot \begin{vmatrix} x & 3 \\ 4 & 5 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 3 \\ 3 & 5 \end{vmatrix} + 3 \cdot \begin{vmatrix} 2 & x \\ 3 & 4 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
Now we will calculate each of the 2x2 determinants:

1. For \(\begin{vmatrix} x & 3 \\ 4 & 5 \end{vmatrix}\):
\[ = x \cdot 5 - 3 \cdot 4 = 5x - 12 \]

2. For \(\begin{vmatrix} 2 & 3 \\ 3 & 5 \end{vmatrix}\):
\[ = 2 \cdot 5 - 3 \cdot 3 = 10 - 9 = 1 \]

3. For \(\begin{vmatrix} 2 & x \\ 3 & 4 \end{vmatrix}\):
\[ = 2 \cdot 4 - 3 \cdot x = 8 - 3x \]

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

\[ D = 1(5x - 12) - 2(1) + 3(8 - 3x) \]

### Step 5: Simplify the expression
Now we simplify:

\[ D = 5x - 12 - 2 + 24 - 9x \]
\[ D = 5x - 9x + 24 - 14 \]
\[ D = -4x + 10 \]

### Step 6: Set the determinant equal to zero
Since we need \(D = 0\):

\[ -4x + 10 = 0 \]

### Step 7: Solve for \(x\)
Now, we solve for \(x\):

\[ -4x = -10 \]
\[ x = \frac{10}{4} = \frac{5}{2} \]

### Final Answer
Thus, the value of \(x\) is:

\[ \boxed{\frac{5}{2}} \]