To solve $x^2 + \frac{1}{x^2}$ given $x + \frac{1}{x} = 5$, we follow these steps:

Step 1: Square both sides of $x + \frac{1}{x} = 5$.

$\left( x + \frac{1}{x} \right)^2 = 5^2$ Expanding the left-hand side: $x^2 + 2 + \frac{1}{x^2} = 25$

Step 2: Subtract 2 from both sides.

$x^2 + \frac{1}{x^2} = 25 - 2$ $x^2 + \frac{1}{x^2} = 23$

Final Answer:

$x^2 + \frac{1}{x^2} = 23$

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Related Questions:

1. What if $x + \frac{1}{x}$ were 7 instead of 5?
2. How do we calculate $x^4 + \frac{1}{x^4}$ from this result?
3. What happens when $x + \frac{1}{x} = -5$?
4. How can we express $x^3 + \frac{1}{x^3}$ from $x + \frac{1}{x} = 5$?
5. How does squaring affect the domain restrictions for $x$?

Tip:

When solving problems like this, squaring both sides is common, but remember it can introduce extraneous solutions. Always verify if needed!