Given that $x + \frac{1}{x} = 25$, we want to find the value of $x^2 + \frac{1}{x^2}$.

Step 1: Square both sides of the equation.

We start by squaring the given equation:

$\left( x + \frac{1}{x} \right)^2 = 25^2$

Expanding the left-hand side:

$x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 625$

Simplifying:

$x^2 + 2 + \frac{1}{x^2} = 625$

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

Now, subtract 2 from both sides:

$x^2 + \frac{1}{x^2} = 625 - 2$

$x^2 + \frac{1}{x^2} = 623$

Thus, the value of $x^2 + \frac{1}{x^2}$ is 623.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What happens when $x + \frac{1}{x}$ takes different values?
2. Can you find $x^3 + \frac{1}{x^3}$ based on this result?
3. How can we factor expressions like $x + \frac{1}{x}$?
4. What if $x + \frac{1}{x} = 0$? What happens to $x^2 + \frac{1}{x^2}$?
5. How does this method apply to more complex fractions or variables?

Tip: Squaring an equation can introduce extra solutions, so always check if they are valid when solving!