Sum of solution of the equation `|{:(1,2,x),(2,3,x^2),(3,5,2):}|=10 ` is :

Options

A. 1
B. `-1`
C. 2
D. 4

Answer

To solve the equation given by the determinant \(|{(1, 2, x), (2, 3, x^2), (3, 5, 2)}| = 10\), we will follow these steps:

### Step 1: Expand the Determinant
We start by expanding the determinant using the first column:

\[ |{(1, 2, x), (2, 3, x^2), (3, 5, 2)}| = 1 \cdot |{(3, x^2), (5, 2)}| - 2 \cdot |{(2, x^2), (3, 2)}| + x \cdot |{(2, 3), (3, 5)}| \]

Calculating each of the 2x2 determinants:

1. \(|{(3, x^2), (5, 2)}| = 3 \cdot 2 - 5 \cdot x^2 = 6 - 5x^2\)
2. \(|{(2, x^2), (3, 2)}| = 2 \cdot 2 - 3 \cdot x^2 = 4 - 3x^2\)
3. \(|{(2, 3), (3, 5)}| = 2 \cdot 5 - 3 \cdot 3 = 10 - 9 = 1\)

Putting it all together:

\[ |{(1, 2, x), (2, 3, x^2), (3, 5, 2)}| = 1(6 - 5x^2) - 2(4 - 3x^2) + x(1) \]

### Step 2: Simplify the Expression
Now, substituting back into the equation:

\[ 6 - 5x^2 - 8 + 6x^2 + x = 10 \]

Combine like terms:

\[ (6 - 8) + (-5x^2 + 6x^2) + x = 10 \]

This simplifies to:

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

### Step 3: Rearranging the Equation
Rearranging gives:

\[ -x^2 + x - 12 = 0 \]

Multiplying through by -1:

\[ x^2 - x + 12 = 0 \]

### Step 4: Factor the Quadratic
Now we need to factor the quadratic equation:

\[ x^2 - x - 12 = 0 \]

We can factor this as:

\[ (x - 4)(x + 3) = 0 \]

### Step 5: Find the Solutions
Setting each factor to zero gives us the solutions:

1. \(x - 4 = 0 \Rightarrow x = 4\)
2. \(x + 3 = 0 \Rightarrow x = -3\)

### Step 6: Sum of the Solutions
Finally, we find the sum of the solutions:

\[ 4 + (-3) = 1 \]

Thus, the sum of the solutions of the equation is:

\[ \boxed{1} \]