If ` x ^(2) - 3x + 1 = 0 ` what is the value of ` ( x^(4) - (1)/( x^(4)))` ?

Options

A. 11
B. 18
C. 47
D. 51

Answer

To solve the equation \( x^2 - 3x + 1 = 0 \) and find the value of \( x^4 - \frac{1}{x^4} \), 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 \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -3, c = 1 \).

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

### Step 2: Find \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). To do this, we can use the identity:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
From the quadratic equation, we can express \( x^2 \) as:
\[ x^2 = 3x - 1 \]
Thus,
\[ x + \frac{1}{x} = \frac{(3x - 1) + 1}{x} = \frac{3x}{x} = 3 \]

### Step 3: Find \( x^2 + \frac{1}{x^2} \)
Using the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting \( x + \frac{1}{x} = 3 \):
\[ x^2 + \frac{1}{x^2} = 3^2 - 2 = 9 - 2 = 7 \]

### Step 4: Find \( x^4 + \frac{1}{x^4} \)
Now we can find \( x^4 + \frac{1}{x^4} \) using the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Substituting \( x^2 + \frac{1}{x^2} = 7 \):
\[ x^4 + \frac{1}{x^4} = 7^2 - 2 = 49 - 2 = 47 \]

### Final Answer
Thus, the value of \( x^4 - \frac{1}{x^4} \) is:
\[ \boxed{47} \]