Expand `|(2,2x),(6,x)|`

Answer

To expand the determinant \( |(2, 2x), (6, x)| \), we can follow these steps:

1. **Set up the determinant**:
We denote the determinant as \( \Delta \):
\[ \Delta = \begin{vmatrix} 2 & 2x \\ 6 & x \end{vmatrix} \]

2. **Apply the formula for a 2x2 determinant**:
The formula for the determinant of a 2x2 matrix \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \) is given by \( ad - bc \).
Here, \( a = 2 \), \( b = 2x \), \( c = 6 \), and \( d = x \).

3. **Calculate the determinant**:
Using the formula:
\[ \Delta = (2)(x) - (6)(2x) \]

4. **Simplify the expression**:
\[ \Delta = 2x - 12x \]

5. **Combine like terms**:
\[ \Delta = -10x \]

Thus, the expanded determinant is:
\[ \Delta = -10x \]