If x is real and `x^(4) - 5x^(2) - 1 = 0`, when the value of `(x^(6) - 3x^(2) + (3)/(x^(2)) - (1)/(x^(6)) + 1)` is :

Options

A. 126
B. 110
C. 116
D. 96

Answer

To solve the equation \( x^4 - 5x^2 - 1 = 0 \) and find the value of \( x^6 - 3x^2 + \frac{3}{x^2} - \frac{1}{x^6} + 1 \), we can follow these steps:

### Step 1: Substitute \( y = x^2 \)
We start by letting \( y = x^2 \). Then the equation becomes:
\[ y^2 - 5y - 1 = 0 \]

### Step 2: Solve the quadratic equation
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -5, c = -1 \):
\[ y = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ y = \frac{5 \pm \sqrt{25 + 4}}{2} \]
\[ y = \frac{5 \pm \sqrt{29}}{2} \]

### Step 3: Find \( x^2 \)
Thus, we have two possible values for \( y \):
\[ y_1 = \frac{5 + \sqrt{29}}{2}, \quad y_2 = \frac{5 - \sqrt{29}}{2} \]
Since \( y = x^2 \), we can take the positive roots since \( x \) is real.

### Step 4: Calculate \( x^6 - \frac{1}{x^6} \)
Next, we need to find \( x^6 - \frac{1}{x^6} \). We can use the identity:
\[ x^6 - \frac{1}{x^6} = (x^2 - \frac{1}{x^2})^3 + 3(x^2 - \frac{1}{x^2}) \]
We first need to find \( x^2 - \frac{1}{x^2} \).

### Step 5: Calculate \( x^2 - \frac{1}{x^2} \)
From the original equation \( x^2 - \frac{1}{x^2} = 5 \), we can cube this:
\[ (x^2 - \frac{1}{x^2})^3 = 5^3 = 125 \]
Now substituting back:
\[ x^6 - \frac{1}{x^6} = 125 + 3(5) = 125 + 15 = 140 \]

### Step 6: Calculate the final expression
Now we need to find:
\[ x^6 - 3x^2 + \frac{3}{x^2} - \frac{1}{x^6} + 1 \]
We already have \( x^6 - \frac{1}{x^6} = 140 \). We also know:
\[ -3x^2 + \frac{3}{x^2} = -3y + \frac{3}{y} \]
Substituting \( y = \frac{5 + \sqrt{29}}{2} \):
\[ -3y + \frac{3}{y} = -3\left(\frac{5 + \sqrt{29}}{2}\right) + \frac{3}{\frac{5 + \sqrt{29}}{2}} = -\frac{15 + 3\sqrt{29}}{2} + \frac{6}{5 + \sqrt{29}} \]
Calculating \( \frac{6}{5 + \sqrt{29}} \) requires rationalizing the denominator.

### Step 7: Combine everything
Finally, we combine:
\[ x^6 - 3x^2 + \frac{3}{x^2} - \frac{1}{x^6} + 1 = 140 - 15 + 1 = 126 \]

Thus, the final value is:
\[ \boxed{126} \]