The sum of the real roots of the equation
`|{:(x, -6, -1), (2, -3x, x-3), (-3, 2x, x+2):}| = 0,` is equal to

Options

A. 0
B. `-4`
C. 6
D. 1

Answer

To solve the equation given by the determinant

\[ \left| \begin{array}{ccc} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{array} \right| = 0, \]

we will follow these steps:

### Step 1: Calculate the determinant

We will expand the determinant along the first row. The determinant can be calculated as follows:

\[ D = x \cdot \left| \begin{array}{cc} -3x & x-3 \\ 2x & x+2 \end{array} \right| - (-6) \cdot \left| \begin{array}{cc} 2 & x-3 \\ -3 & x+2 \end{array} \right| - 1 \cdot \left| \begin{array}{cc} 2 & -3x \\ -3 & 2x \end{array} \right|. \]

### Step 2: Calculate the 2x2 determinants

1. For the first 2x2 determinant:

\[ \left| \begin{array}{cc} -3x & x-3 \\ 2x & x+2 \end{array} \right| = (-3x)(x+2) - (x-3)(2x) = -3x^2 - 6x - 2x^2 + 6x = -5x^2. \]

2. For the second 2x2 determinant:

\[ \left| \begin{array}{cc} 2 & x-3 \\ -3 & x+2 \end{array} \right| = (2)(x+2) - (-3)(x-3) = 2x + 4 + 3x - 9 = 5x - 5. \]

3. For the third 2x2 determinant:

\[ \left| \begin{array}{cc} 2 & -3x \\ -3 & 2x \end{array} \right| = (2)(2x) - (-3)(-3x) = 4x - 9x = -5x. \]

### Step 3: Substitute back into the determinant equation

Now substituting these results back into the determinant expression:

\[ D = x(-5x^2) + 6(5x - 5) - (-5x) = -5x^3 + 30x - 30 + 5x = -5x^3 + 35x - 30. \]

### Step 4: Set the determinant to zero

Now we set the determinant equal to zero:

\[ -5x^3 + 35x - 30 = 0. \]

Dividing the entire equation by -5 gives:

\[ x^3 - 7x + 6 = 0. \]

### Step 5: Find the roots of the cubic equation

To find the roots, we can use the Rational Root Theorem or synthetic division. Testing possible rational roots, we find that \(x = 1\) is a root:

\[ 1^3 - 7(1) + 6 = 0. \]

Now we can factor \(x - 1\) out of the cubic polynomial:

Using synthetic division:

\[ \begin{array}{r|rrrr} 1 & 1 & 0 & -7 & 6 \\ & & 1 & 1 & -6 \\ \hline & 1 & 1 & -6 & 0 \\ \end{array} \]

This gives us:

\[ x^3 - 7x + 6 = (x - 1)(x^2 + x - 6). \]

Now we can factor \(x^2 + x - 6\):

\[ x^2 + x - 6 = (x - 2)(x + 3). \]

Thus, the complete factorization is:

\[ (x - 1)(x - 2)(x + 3) = 0. \]

### Step 6: Find the sum of the real roots

The real roots are \(x = 1\), \(x = 2\), and \(x = -3\). The sum of the real roots is:

\[ 1 + 2 - 3 = 0. \]

### Final Answer

The sum of the real roots of the equation is \(0\).

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