To solve the equation \(33^2 + x^2 = 65^2\) for \(x\), follow these steps:

1. Start by squaring the numbers:
   \[
   33^2 = 1089
   \]
   \[
   65^2 = 4225
   \]

2. Substitute these values into the equation:
   \[
   1089 + x^2 = 4225
   \]

3. Subtract 1089 from both sides of the equation to isolate \(x^2\):
   \[
   x^2 = 4225 - 1089
   \]
   \[
   x^2 = 3136
   \]

4. Finally, take the square root of both sides to solve for \(x\):
   \[
   x = \sqrt{3136}
   \]
   \[
   x = 56
   \]

So, the solution is \(x = 56\).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the square root of a negative number?
2. How do you solve quadratic equations that cannot be factored easily?
3. Can you solve for \(x\) in the equation \(x^2 - 144 = 0\)?
4. How do you solve equations involving cube roots?
5. How can the Pythagorean theorem be used in real-life situations?

**Tip:** Always check the discriminant when solving quadratic equations; it helps you determine the nature of the roots.

Learn how to solve the equation 33^2 + x^2 = 65^2 step-by-step with detailed explanations. This problem involves the Pythagorean theorem and quadratic equations, suitable for high school students.

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