The roots of the equation `|(x,3,7),(2,x,2),(7,6,x)|` = 0 are

Options

A. 9,2,-7
B. 9,-2,7
C. `-9,2,7`
D. none

Answer

To find the roots of the equation given by the determinant \(|(x,3,7),(2,x,2),(7,6,x)| = 0\), we will follow these steps:

### Step 1: Expand the Determinant
We will expand the determinant using the first row. The determinant can be expressed as:

\[ D = x \cdot \begin{vmatrix} x & 2 \\ 6 & x \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 & 2 \\ 7 & x \end{vmatrix} + 7 \cdot \begin{vmatrix} 2 & x \\ 7 & 6 \end{vmatrix} \]

### Step 2: Calculate the 2x2 Determinants
Now, we calculate the 2x2 determinants:

1. \(\begin{vmatrix} x & 2 \\ 6 & x \end{vmatrix} = x^2 - 12\)
2. \(\begin{vmatrix} 2 & 2 \\ 7 & x \end{vmatrix} = 2x - 14\)
3. \(\begin{vmatrix} 2 & x \\ 7 & 6 \end{vmatrix} = 12 - 7x\)

### Step 3: Substitute Back into the Determinant Expression
Substituting these values back into the determinant expression, we have:

\[ D = x(x^2 - 12) - 3(2x - 14) + 7(12 - 7x) \]

### Step 4: Simplify the Expression
Now, we simplify the expression:

\[ D = x^3 - 12x - 6x + 42 + 84 - 49x \]

Combining like terms:

\[ D = x^3 - 67x + 126 \]

### Step 5: Set the Determinant Equal to Zero
We set the determinant equal to zero to find the roots:

\[ x^3 - 67x + 126 = 0 \]

### Step 6: Check Possible Rational Roots
We will check for possible rational roots using the Rational Root Theorem. We can test values such as \(0, 1, -1, 2, -2, 3, -3, 6, -6, 7, -7, 9, -9, 14, -14, 42, -42, 63, -63, 126, -126\).

### Step 7: Testing \(x = 2\)
Testing \(x = 2\):

\[ D(2) = 2^3 - 67(2) + 126 = 8 - 134 + 126 = 0 \]

Thus, \(x = 2\) is a root.

### Step 8: Factor the Polynomial
Since \(x = 2\) is a root, we can factor the polynomial \(x^3 - 67x + 126\) as \((x - 2)(Ax^2 + Bx + C)\).

### Step 9: Perform Polynomial Long Division
We divide \(x^3 - 67x + 126\) by \(x - 2\) to find \(Ax^2 + Bx + C\):

1. Divide \(x^3\) by \(x\) to get \(x^2\).
2. Multiply \(x^2\) by \(x - 2\) to get \(x^3 - 2x^2\).
3. Subtract this from the original polynomial.
4. Repeat this process until you reach a constant.

After performing the division, we find:

\[ x^3 - 67x + 126 = (x - 2)(x^2 + 2x - 63) \]

### Step 10: Solve the Quadratic Equation
Now we solve the quadratic equation \(x^2 + 2x - 63 = 0\) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-63)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ = \frac{-2 \pm \sqrt{4 + 252}}{2} = \frac{-2 \pm \sqrt{256}}{2} = \frac{-2 \pm 16}{2} \]

Thus, we have:

\[ x = \frac{14}{2} = 7 \quad \text{and} \quad x = \frac{-18}{2} = -9 \]

### Final Roots
The roots of the equation are:

\[ x = 2, \quad x = 7, \quad x = -9 \]