The roots of the equation `|(1,4,20),(1,-2,5),(1,2x,5x^(2))| = 0` are

Options

A. `-1, -2`
B. `-1, 2`
C. `1, -2`
D. `1, 2`

Answer

To find the roots of the equation given by the determinant \( |(1, 4, 20), (1, -2, 5), (1, 2x, 5x^2)| = 0 \), we will follow these steps:

### Step 1: Set up the determinant
We start with the determinant:
\[ D = \begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} \]

### Step 2: Apply row operations
We can simplify the determinant using row operations. We will perform the following operations:
- \( R_2 \leftarrow R_2 - R_1 \)
- \( R_3 \leftarrow R_3 - R_1 \)

This gives us:
\[ D = \begin{vmatrix} 1 & 4 & 20 \\ 0 & -6 & -15 \\ 0 & 2x - 4 & 5x^2 - 20 \end{vmatrix} \]

### Step 3: Calculate the determinant
Now, we can calculate the determinant using the first column:
\[ D = 1 \cdot \begin{vmatrix} -6 & -15 \\ 2x - 4 & 5x^2 - 20 \end{vmatrix} \]

Calculating the 2x2 determinant:
\[ D = -6(5x^2 - 20) - (-15)(2x - 4) \]
\[ D = -30x^2 + 120 + 30x - 60 \]
\[ D = -30x^2 + 30x + 60 \]

### Step 4: Factor out the common term
Now, we can factor out \(-30\):
\[ D = -30(x^2 - x - 2) \]

### Step 5: Solve the quadratic equation
Next, we set the determinant equal to zero:
\[ -30(x^2 - x - 2) = 0 \]
This simplifies to:
\[ x^2 - x - 2 = 0 \]

### Step 6: Factor the quadratic
Factoring the quadratic gives us:
\[ (x - 2)(x + 1) = 0 \]

### Step 7: Find the roots
Setting each factor to zero gives us the roots:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Conclusion
The roots of the equation are:
\[ x = 2 \quad \text{and} \quad x = -1 \]