The solution set of the equation `|{:(1,4,20),(1,-2,5),(1,2x,5x^(2)):}|=0` is

Options

A. {1, 2}
B. `{-1,1}`
C. `{-1,2}`
D. none of these

Answer

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

### Step 1: Expand the Determinant
We start by expanding the determinant. The determinant of a 3x3 matrix can be calculated using the formula:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our determinant, we have:

\[ \begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} \]

Using the first row to expand the determinant:

\[ = 1 \cdot \begin{vmatrix} -2 & 5 \\ 2x & 5x^2 \end{vmatrix} - 4 \cdot \begin{vmatrix} 1 & 5 \\ 1 & 5x^2 \end{vmatrix} + 20 \cdot \begin{vmatrix} 1 & -2 \\ 1 & 2x \end{vmatrix} \]

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

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

2. For \( \begin{vmatrix} 1 & 5 \\ 1 & 5x^2 \end{vmatrix} \):
\[ = (1)(5x^2) - (5)(1) = 5x^2 - 5 \]

3. For \( \begin{vmatrix} 1 & -2 \\ 1 & 2x \end{vmatrix} \):
\[ = (1)(2x) - (-2)(1) = 2x + 2 \]

### Step 3: Substitute Back into the Determinant
Now substitute these back into the determinant expansion:

\[ = 1(-10x^2 - 10x) - 4(5x^2 - 5) + 20(2x + 2) \]

### Step 4: Simplify the Expression
Now we simplify the expression:

\[ = -10x^2 - 10x - 20x^2 + 20 + 40x + 40 \]

Combine like terms:

\[ = (-10x^2 - 20x^2) + (-10x + 40x) + (20 + 40) \]
\[ = -30x^2 + 30x + 60 \]

### Step 5: Set the Expression to Zero
Now we set the expression equal to zero:

\[ -30x^2 + 30x + 60 = 0 \]

Dividing the entire equation by -30:

\[ x^2 - x - 2 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we factor the quadratic equation:

\[ (x - 2)(x + 1) = 0 \]

### Step 7: Solve for x
Setting each factor to zero gives us:

1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

### Final Solution
Thus, the solution set of the equation is:

\[ \{ -1, 2 \} \]