To solve this problem, we proceed as follows:

Step 1: Use the given equation

We are given: $x + \frac{1}{x} = 2$

We need to find the value of: $x^2 + \frac{1}{x^2}$

Step 2: Square the given equation

Squaring both sides of $x + \frac{1}{x} = 2$: $\left(x + \frac{1}{x}\right)^2 = 2^2$

Expanding the left-hand side: $x^2 + 2 + \frac{1}{x^2} = 4$

Step 3: Solve for $x^2 + \frac{1}{x^2}$

Rearranging: $x^2 + \frac{1}{x^2} = 4 - 2$

$x^2 + \frac{1}{x^2} = 2$

Final Answer:

$x^2 + \frac{1}{x^2} = 2$

Let me know if you need detailed steps or further assistance!

Related Questions:

1. What happens if $x + \frac{1}{x}$ equals a different number, like $3$?
2. Can you generalize $x^n + \frac{1}{x^n}$ for any $n$?
3. How does squaring affect inequalities in equations like this?
4. What are real-world applications of such transformations?
5. What would the result be if $x + \frac{1}{x}$ were an irrational number?

Tip: Squaring equations is useful but always check for extraneous solutions!