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 need to calculate the determinant and set it equal to zero.

### Step-by-Step Solution:

1. **Write the Determinant**:
The determinant can be expressed as:
\[ D = \begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} \]

2. **Calculate the Determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the second and third rows respectively.

Here, we have:
- \( a = 1, b = 4, c = 20 \)
- \( d = 1, e = -2, f = 5 \)
- \( g = 1, h = 2x, i = 5x^2 \)

The determinant can be calculated as:
\[ D = 1((-2)(5x^2) - (5)(2x)) - 4(1(5x^2) - (5)(1)) + 20(1(2x) - (-2)(1)) \]

Simplifying each term:
- First term: \( 1(-10x^2 - 10x) = -10x^2 - 10x \)
- Second term: \( -4(5x^2 - 5) = -20x^2 + 20 \)
- Third term: \( 20(2x + 2) = 40x + 40 \)

Combining these:
\[ D = -10x^2 - 10x - 20x^2 + 20 + 40x + 40 \]
\[ D = -30x^2 + 30x + 60 \]

3. **Set the Determinant Equal to Zero**:
Now, we set the determinant equal to zero:
\[ -30x^2 + 30x + 60 = 0 \]

4. **Simplify the Equation**:
We can simplify this equation by dividing all terms by -30:
\[ x^2 - x - 2 = 0 \]

5. **Factor the Quadratic**:
We can factor the quadratic:
\[ (x - 2)(x + 1) = 0 \]

6. **Find the Roots**:
Setting each factor to zero gives us the solutions:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Final Solution:
The solution set of the equation is:
\[ \{2, -1\} \]