If `alpha and beta` are the roots of the equations `x^(2) + 6x + 1 = 0`. Then what is `|alpha - beta|` equal to ?
(a)6
(b)12
(c)`3sqrt(2)`
(d)`4sqrt(2)`

Options

A. 6
B. `3sqrt(2)`
C. `4sqrt(2)`
D. 12

Answer

To find \(|\alpha - \beta|\) where \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 + 6x + 1 = 0\), we can use the properties of the roots of a quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients:**
The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 6\), and \(c = 1\).

2. **Calculate the sum and product of the roots:**
The sum of the roots \(\alpha + \beta\) can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{6}{1} = -6 \]
The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{1}{1} = 1 \]

3. **Use the formula for the difference of the roots:**
The difference of the roots can be calculated using the formula:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the values we found:
\[ |\alpha - \beta| = \sqrt{(-6)^2 - 4 \cdot 1} \]

4. **Calculate the expression inside the square root:**
\[ |\alpha - \beta| = \sqrt{36 - 4} = \sqrt{32} \]

5. **Simplify the square root:**
\[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]

6. **Final answer:**
Therefore, \(|\alpha - \beta| = 4\sqrt{2}\).

### Conclusion:
The correct answer is (d) \(4\sqrt{2}\).