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!

Learn how to solve for x^2 + 1/x^2 when x + 1/x = 25 with this step-by-step guide. Explore algebraic concepts such as squaring rational expressions. Suitable for students in Grades 9-11.

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