Sum of real roots of the euation `|{:(1,4,20),(1,-2,5),(1,2x,5x^(2)):}|=0` is

Options

A. -2
B. -1
C. 0
D. 1

Answer

To solve the given problem, we need to evaluate the determinant and find the sum of its real roots. The determinant is given as:

\[ \begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} = 0 \]

### Step 1: Calculate the Determinant
We can expand the determinant along the first column:

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

### Step 2: Calculate the 2x2 Determinants
1. **First determinant**:
\[ \begin{vmatrix} -2 & 5 \\ 2x & 5x^2 \end{vmatrix} = (-2)(5x^2) - (5)(2x) = -10x^2 - 10x \]

2. **Second determinant**:
\[ \begin{vmatrix} 4 & 20 \\ 2x & 5x^2 \end{vmatrix} = (4)(5x^2) - (20)(2x) = 20x^2 - 40x \]

3. **Third determinant**:
\[ \begin{vmatrix} 4 & 20 \\ -2 & 5 \end{vmatrix} = (4)(5) - (20)(-2) = 20 + 40 = 60 \]

### Step 3: Substitute Back into the Determinant
Now substituting these values back into the determinant expression:

\[ D = 1(-10x^2 - 10x) - 1(20x^2 - 40x) + 1(60) \]

This simplifies to:

\[ D = -10x^2 - 10x - 20x^2 + 40x + 60 \]

Combining like terms:

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

### Step 4: Set the Determinant to Zero
Now we set the determinant equal to zero:

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

Dividing through by -30 gives:

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

### Step 5: Solve the Quadratic Equation
We can factor this quadratic equation:

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

Thus, the roots are:

\[ x = 2 \quad \text{and} \quad x = -1 \]

### Step 6: Find the Sum of the Real Roots
The sum of the real roots is:

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

### Final Answer
The sum of the real roots of the equation is:

\[ \boxed{1} \]