If `(x+1/x)=sqrt(7)`, then `x^(2) + 1/x^(2)` = _______

Options

A. 5
B. 4
C. 0
D. 3

Answer

To solve the problem, we start with the given equation:

\[ x + \frac{1}{x} = \sqrt{7} \]

We want to find the value of:

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

### Step 1: Square both sides of the equation

We will square both sides of the equation \(x + \frac{1}{x} = \sqrt{7}\):

\[ \left(x + \frac{1}{x}\right)^2 = (\sqrt{7})^2 \]

### Step 2: Expand the left side using the formula for squaring a binomial

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we expand the left side:

\[ x^2 + 2\left(x \cdot \frac{1}{x}\right) + \frac{1}{x^2} = 7 \]

### Step 3: Simplify the equation

Since \(x \cdot \frac{1}{x} = 1\), we can simplify the equation:

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

### Step 4: Isolate \(x^2 + \frac{1}{x^2}\)

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

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

### Step 5: Calculate the final result

So we have:

\[ x^2 + \frac{1}{x^2} = 5 \]

Thus, the final answer is:

\[ \boxed{5} \]