If ` x + (1 / x)`= `(sqrt(3) + 1)/ 2` then what is the value of ` x^(4) + 1 / x^(4)` ?

Options

A. `( 4 sqrt"'3 - 1) // 4`
B. `( 4 sqrt""3 + 1 ) // 2`
C. `(-4 sqrt""3 - 1) // 4`
D. `(-4 sqrt""3 - 1 ) // 2 `

Answer

To find the value of \( x^4 + \frac{1}{x^4} \) given that \( x + \frac{1}{x} = \frac{\sqrt{3} + 1}{2} \), we can follow these steps:

### Step 1: Square the given equation
Start by squaring both sides of the equation:
\[ \left( x + \frac{1}{x} \right)^2 = \left( \frac{\sqrt{3} + 1}{2} \right)^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = \frac{(\sqrt{3} + 1)^2}{4} \]

### Step 2: Calculate the right-hand side
Now calculate \( (\sqrt{3} + 1)^2 \):
\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
Thus, we have:
\[ x^2 + 2 + \frac{1}{x^2} = \frac{4 + 2\sqrt{3}}{4} \]

### Step 3: Simplify the equation
This simplifies to:
\[ x^2 + \frac{1}{x^2} + 2 = 1 + \frac{\sqrt{3}}{2} \]
Subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 1 + \frac{\sqrt{3}}{2} - 2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = \frac{\sqrt{3}}{2} - 1 \]

### Step 4: Square again to find \( x^4 + \frac{1}{x^4} \)
Now square \( x^2 + \frac{1}{x^2} \):
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = \left( \frac{\sqrt{3}}{2} - 1 \right)^2 \]
This expands to:
\[ x^4 + 2 + \frac{1}{x^4} = \left( \frac{\sqrt{3}}{2} - 1 \right)^2 \]

### Step 5: Calculate the right-hand side
Calculate \( \left( \frac{\sqrt{3}}{2} - 1 \right)^2 \):
\[ \left( \frac{\sqrt{3}}{2} - 1 \right)^2 = \left( \frac{\sqrt{3}}{2} - \frac{2}{2} \right)^2 = \left( \frac{\sqrt{3} - 2}{2} \right)^2 = \frac{(\sqrt{3} - 2)^2}{4} \]
Expanding \( (\sqrt{3} - 2)^2 \):
\[ (\sqrt{3} - 2)^2 = 3 - 4\sqrt{3} + 4 = 7 - 4\sqrt{3} \]
Thus, we have:
\[ x^4 + 2 + \frac{1}{x^4} = \frac{7 - 4\sqrt{3}}{4} \]

### Step 6: Solve for \( x^4 + \frac{1}{x^4} \)
Subtract 2 from both sides:
\[ x^4 + \frac{1}{x^4} = \frac{7 - 4\sqrt{3}}{4} - 2 \]
Convert 2 into a fraction with a denominator of 4:
\[ 2 = \frac{8}{4} \]
Thus:
\[ x^4 + \frac{1}{x^4} = \frac{7 - 4\sqrt{3} - 8}{4} = \frac{-1 - 4\sqrt{3}}{4} \]

### Final Answer
Therefore, the value of \( x^4 + \frac{1}{x^4} \) is:
\[ \boxed{\frac{-1 - 4\sqrt{3}}{4}} \]