If `x=-9` is a root of `|(x,3,7),(2,x,2),(7,6,x)|=0` then other two roots are…………………

Explanation

To solve the problem, we need to find the other two roots of the determinant equation given that one root is \( x = -9 \). The determinant is given as:

\[ \begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0 \]

### Step 1: Calculate the Determinant

We will expand the determinant along the first row:

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

Calculating each of 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. \( \begin{vmatrix} 2 & x \\ 7 & 6 \end{vmatrix} = 12 - 7x \)

Substituting these back into the determinant expression:

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

### Step 2: Simplify the Determinant

Now, we simplify \( D \):

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

Combining like terms:

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

### Step 3: Set the Determinant to Zero

Since we know that \( x = -9 \) is a root, we can substitute \( x = -9 \) into the equation:

\[ (-9)^3 - 67(-9) + 126 = 0 \]

Calculating this:

\[ -729 + 603 + 126 = 0 \]

This confirms that \( x = -9 \) is indeed a root.

### Step 4: Use Vieta's Formulas

Let the other two roots be \( \alpha \) and \( \beta \). According to Vieta's formulas:

1. The sum of the roots \( \alpha + \beta + (-9) = 0 \) implies \( \alpha + \beta = 9 \).
2. The product of the roots \( \alpha \beta (-9) = -126 \) implies \( \alpha \beta = 14 \).

### Step 5: Solve for \( \alpha \) and \( \beta \)

Now we have the system of equations:

1. \( \alpha + \beta = 9 \)
2. \( \alpha \beta = 14 \)

We can express \( \beta \) in terms of \( \alpha \):

\[ \beta = 9 - \alpha \]

Substituting into the product equation:

\[ \alpha(9 - \alpha) = 14 \]

Expanding and rearranging gives us:

\[ \alpha^2 - 9\alpha + 14 = 0 \]

### Step 6: Solve the Quadratic Equation

Using the quadratic formula:

\[ \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = -9, c = 14 \):

\[ \alpha = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 14}}{2 \cdot 1} \]
\[ = \frac{9 \pm \sqrt{81 - 56}}{2} \]
\[ = \frac{9 \pm \sqrt{25}}{2} \]
\[ = \frac{9 \pm 5}{2} \]

Calculating the two possible values:

1. \( \alpha = \frac{14}{2} = 7 \)
2. \( \beta = \frac{4}{2} = 2 \)

Thus, the other two roots are \( \alpha = 7 \) and \( \beta = 2 \).

### Final Answer

The other two roots are \( 7 \) and \( 2 \).

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