If `x^(2) - 3x - 1 = 0`, then the value of `(x^(2) + 8x - 1)(x^(3) + x^(-1))^(-1)` is

Options

A. `(3)/(8)`
B. 8
C. 1
D. 3

Answer

To solve the equation \( x^2 - 3x - 1 = 0 \) and find the value of \( (x^2 + 8x - 1)(x^3 + x^{-1})^{-1} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the equation:
\[ x^2 - 3x - 1 = 0 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -3, c = -1 \). Plugging in these values:
\[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{3 \pm \sqrt{9 + 4}}{2} \]
\[ x = \frac{3 \pm \sqrt{13}}{2} \]

### Step 2: Substitute \( x \) into the expression
We need to evaluate:
\[ (x^2 + 8x - 1)(x^3 + x^{-1})^{-1} \]
First, calculate \( x^2 + 8x - 1 \):
Using \( x^2 = 3x + 1 \) (from the original equation):
\[ x^2 + 8x - 1 = (3x + 1) + 8x - 1 = 11x \]

### Step 3: Calculate \( x^3 + x^{-1} \)
To find \( x^3 \), we can use \( x^3 = x \cdot x^2 \):
\[ x^3 = x(3x + 1) = 3x^2 + x \]
Substituting \( x^2 = 3x + 1 \):
\[ x^3 = 3(3x + 1) + x = 9x + 3 + x = 10x + 3 \]
Now, we need \( x^{-1} \):
From the original equation \( x^2 - 3x - 1 = 0 \), we can express \( x^{-1} \) as:
\[ x^{-1} = \frac{1}{x} = \frac{3 - x}{1} \text{ (from rearranging the equation)} \]

Thus, \( x^3 + x^{-1} \) becomes:
\[ x^3 + x^{-1} = (10x + 3) + \frac{3 - x}{1} = 10x + 3 + 3 - x = 9x + 6 \]

### Step 4: Evaluate the entire expression
Now we substitute back into the expression:
\[ (11x)(9x + 6)^{-1} \]
This simplifies to:
\[ \frac{11x}{9x + 6} \]

### Step 5: Simplify further
Factoring out \( 3 \) from the denominator:
\[ \frac{11x}{3(3x + 2)} = \frac{11}{3} \cdot \frac{x}{3x + 2} \]

### Final Step: Substitute \( x \)
Now we can substitute \( x = \frac{3 + \sqrt{13}}{2} \) or \( x = \frac{3 - \sqrt{13}}{2} \) into the expression, but we can also evaluate the expression directly since it is independent of the specific value of \( x \).

### Conclusion
Thus, the final value of \( (x^2 + 8x - 1)(x^3 + x^{-1})^{-1} \) is:
\[ \frac{11}{3} \]