Given that \( x + x^{-1} = 37 \), we want to find the value of \( x^2 + x^{-2} \).

Here's how we can solve this:

1. Start with the given equation:
   \[
   x + x^{-1} = 37
   \]

2. Square both sides of the equation to find a relation for \( x^2 + x^{-2} \):
   \[
   (x + x^{-1})^2 = 37^2
   \]

3. Expand the left-hand side:
   \[
   x^2 + 2xx^{-1} + x^{-2} = 1369
   \]
   Note that \( xx^{-1} = 1 \), so:
   \[
   x^2 + 2 + x^{-2} = 1369
   \]

4. Subtract 2 from both sides to isolate \( x^2 + x^{-2} \):
   \[
   x^2 + x^{-2} = 1369 - 2
   \]
   \[
   x^2 + x^{-2} = 1367
   \]

Therefore, the value of \( x^2 + x^{-2} \) is \( \boxed{1367} \).

Would you like more details or have any questions? Here are some related questions to further explore:

1. What is the value of \( x^3 + x^{-3} \) given \( x + x^{-1} = 37 \)?
2. How do you solve for \( x \) given \( x + x^{-1} = 37 \)?
3. If \( x + x^{-1} = k \), can you derive a general formula for \( x^2 + x^{-2} \) in terms of \( k \)?
4. What happens to the value of \( x^2 + x^{-2} \) if \( x + x^{-1} \) is negative?
5. How do these equations change if we work with complex numbers instead of real numbers?

**Tip:** Squaring or cubing equations is a common technique to derive new relationships, especially when dealing with reciprocal terms.

Learn how to solve for x^2 + x^-2 when given the equation x + x^-1 = 37. Step-by-step solution provided using reciprocal identities and quadratic equation solving techniques. Suitable for high school students studying algebra and quadratic equations.

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