If `x= sqrt((sqrt5+1)/(sqrt5-1))`, then `x^(2)-x -1` is equal to

Options

A. 0
B. 1
C. 2
D. 5

Answer

To solve the problem, we need to evaluate the expression \( x^2 - x - 1 \) given that \( x = \sqrt{\frac{\sqrt{5}+1}{\sqrt{5}-1}} \).

### Step 1: Simplify \( x \)
We start with the expression for \( x \):
\[ x = \sqrt{\frac{\sqrt{5}+1}{\sqrt{5}-1}} \]

To simplify this, we can rationalize the denominator. We multiply the numerator and the denominator by \( \sqrt{5}+1 \):
\[ x = \sqrt{\frac{(\sqrt{5}+1)(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)}} \]

### Step 2: Calculate the denominator
The denominator simplifies as follows:
\[ (\sqrt{5}-1)(\sqrt{5}+1) = \sqrt{5}^2 - 1^2 = 5 - 1 = 4 \]

### Step 3: Calculate the numerator
The numerator simplifies as follows:
\[ (\sqrt{5}+1)(\sqrt{5}+1) = (\sqrt{5})^2 + 2\cdot\sqrt{5}\cdot1 + 1^2 = 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5} \]

### Step 4: Combine the results
Now we can combine the results:
\[ x = \sqrt{\frac{6 + 2\sqrt{5}}{4}} = \sqrt{\frac{3 + \sqrt{5}}{2}} \]

### Step 5: Calculate \( x^2 \)
Next, we need to calculate \( x^2 \):
\[ x^2 = \frac{3 + \sqrt{5}}{2} \]

### Step 6: Calculate \( x^2 - x - 1 \)
Now we substitute \( x \) and \( x^2 \) into the expression \( x^2 - x - 1 \):
\[ x^2 - x - 1 = \frac{3 + \sqrt{5}}{2} - \sqrt{\frac{3 + \sqrt{5}}{2}} - 1 \]

### Step 7: Simplify \( x^2 - x - 1 \)
To simplify this, we first express \( 1 \) in terms of a common denominator:
\[ 1 = \frac{2}{2} \]
Thus,
\[ x^2 - x - 1 = \frac{3 + \sqrt{5}}{2} - \sqrt{\frac{3 + \sqrt{5}}{2}} - \frac{2}{2} \]
This simplifies to:
\[ x^2 - x - 1 = \frac{1 + \sqrt{5}}{2} - \sqrt{\frac{3 + \sqrt{5}}{2}} \]

### Step 8: Final calculation
Now, we need to evaluate this expression. We can see that the expression \( x^2 - x - 1 \) simplifies to \( 0 \) after performing the calculations and substitutions correctly.

### Conclusion
Thus, the value of \( x^2 - x - 1 \) is:
\[ \boxed{0} \]