If `x ^(2) - 3x + 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 for \( x + \frac{1}{x} \)

From the equation \( x^2 - 3x + 1 = 0 \), we can rewrite it as:

\[ x^2 = 3x - 1 \]

Now, divide the entire equation by \( x \) (assuming \( x \neq 0 \)):

\[ x - 3 + \frac{1}{x} = 0 \]

Rearranging gives:

\[ x + \frac{1}{x} = 3 \]

### Step 2: Square \( x + \frac{1}{x} \)

Now, we will square both sides:

\[ \left( x + \frac{1}{x} \right)^2 = 3^2 \]

This expands to:

\[ x^2 + 2 + \frac{1}{x^2} = 9 \]

### Step 3: Isolate \( x^2 + \frac{1}{x^2} \)

Now, we can isolate \( x^2 + \frac{1}{x^2} \):

\[ x^2 + \frac{1}{x^2} = 9 - 2 \]

Thus,

\[ x^2 + \frac{1}{x^2} = 7 \]

### Final Answer

The value of \( x^2 + \frac{1}{x^2} \) is \( 7 \).

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