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: Write the determinant and set it equal to 10
We start with the determinant:
\[ D = \begin{vmatrix} 1 & 2 & x \\ 2 & 3 & x^2 \\ 3 & 5 & 2 \end{vmatrix} \]
We set this equal to 10:
\[ D = 10 \]

### Step 2: Expand the determinant
We can expand the determinant using the first row:
\[ D = 1 \cdot \begin{vmatrix} 3 & x^2 \\ 5 & 2 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & x^2 \\ 3 & 2 \end{vmatrix} + x \cdot \begin{vmatrix} 2 & 3 \\ 3 & 5 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \(\begin{vmatrix} 3 & x^2 \\ 5 & 2 \end{vmatrix} = (3 \cdot 2) - (5 \cdot x^2) = 6 - 5x^2\)
2. \(\begin{vmatrix} 2 & x^2 \\ 3 & 2 \end{vmatrix} = (2 \cdot 2) - (3 \cdot x^2) = 4 - 3x^2\)
3. \(\begin{vmatrix} 2 & 3 \\ 3 & 5 \end{vmatrix} = (2 \cdot 5) - (3 \cdot 3) = 10 - 9 = 1\)

Substituting these back into the determinant:
\[ D = 1(6 - 5x^2) - 2(4 - 3x^2) + x(1) \]
Simplifying this gives:
\[ D = 6 - 5x^2 - 8 + 6x^2 + x = 6 - 8 + x + (6x^2 - 5x^2) \]
\[ D = -2 + x + x^2 \]

### Step 3: Set the determinant equal to 10
Now we set the expanded determinant equal to 10:
\[ x^2 + x - 2 = 10 \]
Rearranging gives:
\[ x^2 + x - 12 = 0 \]

### Step 4: Factor the quadratic equation
We can factor the quadratic equation:
\[ x^2 + 4x - 3x - 12 = 0 \]
This can be factored as:
\[ (x - 3)(x + 4) = 0 \]

### Step 5: Solve for x
Setting each factor to zero gives us the solutions:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 4 = 0 \Rightarrow x = -4\)

### Step 6: Find the sum of the solutions
The sum of the solutions is:
\[ 3 + (-4) = -1 \]

### Final Answer
The sum of the solutions of the equation is \(-1\).

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