The roots of the equation `|{:(x,alpha,1),(beta,x,1),(beta,gamma,1):}|=0` are independent of

Options

A. `alpha`
B. `beta`
C. `gamma`
D. `alpha, beta and gamma`

Explanation

To solve the problem, we need to analyze the determinant of the given matrix and find the roots of the equation formed by setting the determinant equal to zero. The matrix is:

\[ \begin{vmatrix} x & \alpha & 1 \\ \beta & x & 1 \\ \beta & \gamma & 1 \end{vmatrix} \]

### Step 1: Calculate the determinant

We will calculate the determinant of the matrix using the formula for a 3x3 determinant:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

- \( a = x \), \( b = \alpha \), \( c = 1 \)
- \( d = \beta \), \( e = x \), \( f = 1 \)
- \( g = \beta \), \( h = \gamma \), \( i = 1 \)

Thus, the determinant can be calculated as follows:

\[ \text{det}(A) = x \cdot (x \cdot 1 - 1 \cdot \gamma) - \alpha \cdot (\beta \cdot 1 - 1 \cdot \beta) + 1 \cdot (\beta \cdot \gamma - x \cdot \beta) \]

This simplifies to:

\[ \text{det}(A) = x(x - \gamma) - \alpha(0) + (\beta \gamma - \beta x) \]

So we have:

\[ \text{det}(A) = x^2 - x\gamma + \beta \gamma - \beta x \]

### Step 2: Rearranging the determinant expression

Rearranging the terms gives us:

\[ \text{det}(A) = x^2 - x(\gamma + \beta) + \beta \gamma \]

### Step 3: Set the determinant to zero

Now, we set the determinant equal to zero to find the roots:

\[ x^2 - x(\gamma + \beta) + \beta \gamma = 0 \]

### Step 4: Use the quadratic formula

We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots. Here, \( a = 1 \), \( b = -(\gamma + \beta) \), and \( c = \beta \gamma \).

Substituting these values into the quadratic formula gives:

\[ x = \frac{(\gamma + \beta) \pm \sqrt{(\gamma + \beta)^2 - 4 \cdot 1 \cdot \beta \gamma}}{2 \cdot 1} \]

### Step 5: Simplify the expression under the square root

Calculating the discriminant:

\[ (\gamma + \beta)^2 - 4\beta \gamma = \gamma^2 + 2\beta\gamma + \beta^2 - 4\beta\gamma = \gamma^2 - 2\beta\gamma + \beta^2 = (\gamma - \beta)^2 \]

### Step 6: Substitute back into the formula

Now substituting back, we have:

\[ x = \frac{(\gamma + \beta) \pm (\gamma - \beta)}{2} \]

This yields two roots:

1. \( x = \frac{2\gamma}{2} = \gamma \)
2. \( x = \frac{2\beta}{2} = \beta \)

### Conclusion

The roots of the equation are \( \beta \) and \( \gamma \). The problem asks for the variable that the roots are independent of. Since \( \alpha \) does not appear in the roots, we conclude that the roots are independent of \( \alpha \).

### Final Answer

The roots of the equation are independent of \( \alpha \).

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