If `x^(2)-3x+1=0` , then one of the values of `(x^(2)-8x-1)/(x^(3)+1//x)` is :

Options

A. `(sqrt(5)+8)/(7)`
B. `-7//3`
C. `-3`
D. `(sqrt(5)-8)/(7)`

Answer

To solve the problem, we need to find the value of \(\frac{x^2 - 8x - 1}{x^3 + \frac{1}{x}}\) given that \(x^2 - 3x + 1 = 0\).

### Step 1: Solve the quadratic equation
First, we need to find the roots of the equation \(x^2 - 3x + 1 = 0\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -3\), and \(c = 1\).

Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]
Now substituting into the quadratic formula:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 2: Calculate \(x^2\) and \(\frac{1}{x}\)
Next, we need to find \(x^2\) and \(\frac{1}{x}\) for both roots. We can use the quadratic equation to express \(x^2\):
From \(x^2 - 3x + 1 = 0\), we have:
\[ x^2 = 3x - 1 \]

Now, we can find \(\frac{1}{x}\):
Using the roots, we can express \(\frac{1}{x}\) as:
\[ \frac{1}{x} = \frac{2}{3 \pm \sqrt{5}} \]

### Step 3: Substitute into the expression
Now we substitute \(x^2\) into the expression \(\frac{x^2 - 8x - 1}{x^3 + \frac{1}{x}}\).

Calculating \(x^3\):
\[ x^3 = x \cdot x^2 = x(3x - 1) = 3x^2 - x = 3(3x - 1) - x = 9x - 3 - x = 8x - 3 \]

Now, substituting into the expression:
\[ x^3 + \frac{1}{x} = (8x - 3) + \frac{1}{x} \]

### Step 4: Simplify the expression
Now we simplify:
\[ \frac{x^2 - 8x - 1}{x^3 + \frac{1}{x}} = \frac{(3x - 1) - 8x - 1}{(8x - 3) + \frac{1}{x}} = \frac{-5x - 2}{8x - 3 + \frac{1}{x}} \]

### Step 5: Find a common denominator
To simplify further, we can find a common denominator for the denominator:
\[ 8x - 3 + \frac{1}{x} = \frac{(8x^2 - 3x + 1)}{x} \]

### Step 6: Final simplification
Thus, we can write:
\[ \frac{-5x - 2}{\frac{8x^2 - 3x + 1}{x}} = \frac{-5x - 2}{8x^2 - 3x + 1} \cdot x \]

### Step 7: Evaluate the expression
Now we can evaluate this expression using the roots we found earlier.

### Conclusion
After substituting and simplifying, we find that one of the values of the expression is:
\[ \frac{\sqrt{5} - 8}{7} \]