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

Options

A. 3
B. 7
C. 9
D. 11

Answer

To solve the equation \( x^2 - 3x + 1 = 0 \) and find the value of \( x^2 + \frac{1}{x^2} \), 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, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -3 \), and \( c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]

### Step 3: Substitute into the quadratic formula
Now, substituting the values into the quadratic formula:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 4: Find \( x^2 \)
Next, we need to find \( x^2 \). We can use the fact that from the original equation \( x^2 = 3x - 1 \).

### Step 5: Calculate \( \frac{1}{x^2} \)
To find \( \frac{1}{x^2} \), we can use the identity:
\[ \frac{1}{x^2} = \frac{1}{3x - 1} \]

### Step 6: Find \( x^2 + \frac{1}{x^2} \)
Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (3x - 1) + \frac{1}{3x - 1} \]

### Step 7: Simplify the expression
To simplify \( x^2 + \frac{1}{x^2} \), we can find a common denominator:
\[ x^2 + \frac{1}{x^2} = \frac{(3x - 1)(3x - 1) + 1}{3x - 1} \]
Calculating the numerator:
\[ (3x - 1)^2 + 1 = 9x^2 - 6x + 1 + 1 = 9x^2 - 6x + 2 \]
Thus,
\[ x^2 + \frac{1}{x^2} = \frac{9x^2 - 6x + 2}{3x - 1} \]

### Step 8: Substitute \( x^2 \) back into the equation
Now we can substitute \( x^2 = 3x - 1 \):
\[ x^2 + \frac{1}{x^2} = \frac{9(3x - 1) - 6x + 2}{3x - 1} \]
Calculating the numerator:
\[ 27x - 9 - 6x + 2 = 21x - 7 \]
Thus,
\[ x^2 + \frac{1}{x^2} = \frac{21x - 7}{3x - 1} \]

### Final Result
Now we can evaluate this expression for \( x = \frac{3 + \sqrt{5}}{2} \) or \( x = \frac{3 - \sqrt{5}}{2} \) to find the numerical value.