If `|(4x,6x+2,8x+1),(6x+2,9x+3,12x),(8x+1,12x,16x+2)|` = 0 then x =

Options

A. 0
B. `-11`
C. 97
D. `-11//97`

Answer

To solve the determinant equation \( |(4x, 6x+2, 8x+1), (6x+2, 9x+3, 12x), (8x+1, 12x, 16x+2)| = 0 \), we will follow these steps:

### Step 1: Set Up the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} 4x & 6x + 2 & 8x + 1 \\ 6x + 2 & 9x + 3 & 12x \\ 8x + 1 & 12x & 16x + 2 \end{vmatrix} \]

### Step 2: Row Operations
To simplify the determinant, we can perform row operations. We will modify the second and third rows to eliminate the \(x\) terms.

1. **Row 2**: Replace \( R_2 \) with \( R_2 - \frac{3}{2} R_1 \)
2. **Row 3**: Replace \( R_3 \) with \( R_3 - 2 R_1 \)

After performing these operations, we calculate the new rows:
- For \( R_2 \):
\[ R_2 = (6x + 2 - \frac{3}{2} \cdot 4x, 9x + 3 - \frac{3}{2} \cdot (6x + 2), 12x - \frac{3}{2} \cdot (8x + 1)) \]
Simplifying gives:
\[ R_2 = (0, 0, 0) \]

- For \( R_3 \):
\[ R_3 = (8x + 1 - 2 \cdot 4x, 12x - 2 \cdot (6x + 2), 16x + 2 - 2 \cdot (8x + 1)) \]
Simplifying gives:
\[ R_3 = (0, -4, 0) \]

Now the determinant looks like:
\[ D = \begin{vmatrix} 4x & 6x + 2 & 8x + 1 \\ 0 & 0 & 0 \\ 0 & -4 & 0 \end{vmatrix} \]

### Step 3: Calculate the Determinant
The determinant of a matrix with a row of zeros is zero. Therefore, we have:
\[ D = 0 \]

### Step 4: Solve for \(x\)
Since the determinant is equal to zero, we need to find the values of \(x\) that satisfy the original determinant being zero. We can analyze the conditions under which the rows are linearly dependent.

From our operations, we can see that the first row is dependent on the second and third rows. This leads us to set up the equations based on the coefficients of \(x\):
1. From the first row, we have the coefficients of \(x\):
- \(4x\)
- \(6x + 2\)
- \(8x + 1\)

Setting the determinant to zero gives us the condition for linear dependence.

### Step 5: Solve the Equation
To find \(x\), we can set up the equation based on the coefficients:
\[ -8(8x + 1) + 3(6x + 2) = 0 \]
Expanding and simplifying:
\[ -64x - 8 + 18x + 6 = 0 \\ -46x - 2 = 0 \\ 46x = -2 \\ x = -\frac{2}{46} = -\frac{1}{23} \]

### Final Answer
Thus, the value of \(x\) is:
\[ x = -\frac{1}{23} \]