The solution of the equation `|(x,2,-1),(2,5,x),(-1,2,x)|` = 0 are

Options

A. `3,-1`
B. `-3,1`
C. `3,1`
D. `-3,-1`

Answer

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

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} x & 2 & -1 \\ 2 & 5 & x \\ -1 & 2 & x \end{vmatrix} \]

### Step 2: Expand the Determinant
We will use the first row to expand the determinant:
\[ D = x \begin{vmatrix} 5 & x \\ 2 & x \end{vmatrix} - 2 \begin{vmatrix} 2 & x \\ -1 & x \end{vmatrix} - 1 \begin{vmatrix} 2 & 5 \\ -1 & 2 \end{vmatrix} \]

### Step 3: Calculate the 2x2 Determinants
Now we calculate each of the 2x2 determinants:

1. For \( \begin{vmatrix} 5 & x \\ 2 & x \end{vmatrix} \):
\[ = (5)(x) - (2)(x) = 5x - 2x = 3x \]

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

3. For \( \begin{vmatrix} 2 & 5 \\ -1 & 2 \end{vmatrix} \):
\[ = (2)(2) - (5)(-1) = 4 + 5 = 9 \]

### Step 4: Substitute Back into the Determinant
Substituting these values back into the determinant:
\[ D = x(3x) - 2(3x) - 1(9) \]
\[ D = 3x^2 - 6x - 9 \]

### Step 5: Set the Determinant Equal to Zero
Now we set the determinant equal to zero:
\[ 3x^2 - 6x - 9 = 0 \]

### Step 6: Simplify the Equation
Divide the entire equation by 3:
\[ x^2 - 2x - 3 = 0 \]

### Step 7: Factor the Quadratic Equation
Next, we factor the quadratic:
\[ (x - 3)(x + 1) = 0 \]

### Step 8: Solve for x
Setting each factor to zero gives us:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

### Final Solution
Thus, the solutions for the equation \( |(x, 2, -1), (2, 5, x), (-1, 2, x)| = 0 \) are:
\[ x = -1 \quad \text{and} \quad x = 3 \]