If `x ^(2) -x sqrt 68 + 1=0` then
what is the value of `x - (1)/(x)` ?

Options

A. `sqrt66`
B. 8
C. `sqrt62`
D. 6

Answer

To solve the equation \( x^2 - x\sqrt{68} + 1 = 0 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ x^2 - x\sqrt{68} + 1 = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -\sqrt{68}, c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-\sqrt{68})^2 = 68 \]
\[ 4ac = 4 \cdot 1 \cdot 1 = 4 \]
Thus, the discriminant is:
\[ 68 - 4 = 64 \]

### Step 3: Find the roots using the quadratic formula
Now we can substitute the values into the quadratic formula:
\[ x = \frac{\sqrt{68} \pm \sqrt{64}}{2} \]
Since \( \sqrt{64} = 8 \), we have:
\[ x = \frac{\sqrt{68} \pm 8}{2} \]

### Step 4: Simplify the roots
We can express \( \sqrt{68} \) as \( 2\sqrt{17} \):
\[ x = \frac{2\sqrt{17} \pm 8}{2} \]
This simplifies to:
\[ x = \sqrt{17} \pm 4 \]
So, the two possible values for \( x \) are:
\[ x_1 = \sqrt{17} + 4 \quad \text{and} \quad x_2 = \sqrt{17} - 4 \]

### Step 5: Calculate \( \frac{1}{x} \)
Now we need to find \( \frac{1}{x} \) for both cases.

1. For \( x_1 = \sqrt{17} + 4 \):
\[ \frac{1}{x_1} = \frac{1}{\sqrt{17} + 4} \]
To rationalize the denominator:
\[ \frac{1}{x_1} = \frac{\sqrt{17} - 4}{(\sqrt{17} + 4)(\sqrt{17} - 4)} = \frac{\sqrt{17} - 4}{17 - 16} = \sqrt{17} - 4 \]

2. For \( x_2 = \sqrt{17} - 4 \):
\[ \frac{1}{x_2} = \frac{1}{\sqrt{17} - 4} \]
Rationalizing the denominator:
\[ \frac{1}{x_2} = \frac{\sqrt{17} + 4}{(\sqrt{17} - 4)(\sqrt{17} + 4)} = \frac{\sqrt{17} + 4}{17 - 16} = \sqrt{17} + 4 \]

### Step 6: Calculate \( x - \frac{1}{x} \)
Now we can find \( x - \frac{1}{x} \) for both cases.

1. For \( x_1 = \sqrt{17} + 4 \):
\[ x_1 - \frac{1}{x_1} = (\sqrt{17} + 4) - (\sqrt{17} - 4) = 8 \]

2. For \( x_2 = \sqrt{17} - 4 \):
\[ x_2 - \frac{1}{x_2} = (\sqrt{17} - 4) - (\sqrt{17} + 4) = -8 \]

### Conclusion
Thus, the value of \( x - \frac{1}{x} \) can be either \( 8 \) or \( -8 \). However, since we are looking for the positive value, the answer is:
\[ \boxed{8} \]