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To solve the equation \( x^2 = -25 \) over the complex numbers, we follow these steps:

1. Rewrite the equation:
   \[
   x^2 = -25
   \]

2. Take the square root of both sides:
   \[
   x = \pm \sqrt{-25}
   \]

3. Recall that \(\sqrt{-25}\) can be expressed as \(\sqrt{25} \cdot \sqrt{-1}\). Using \(i = \sqrt{-1}\), we get:
   \[
   \sqrt{-25} = 5i
   \]

4. Therefore, the solutions are:
   \[
   x = \pm 5i
   \]

The two solutions are:
\[
x = 5i \quad \text{and} \quad x = -5i
\]

### Correct Answers:
- \( 5i \)
- \( -5i \)

Let me know if you'd like further explanation or have any questions. Here are 5 related questions to expand your understanding:

1. What is the general process for solving equations involving complex numbers?
2. How do square roots of negative numbers work in terms of imaginary numbers?
3. Why is \(i^2 = -1\) an essential property in complex numbers?
4. Can you solve similar equations like \(x^2 = -49\) or \(x^2 = -4\)?
5. What is the geometric interpretation of solutions like \( \pm 5i \) on the complex plane?

**Tip:** When solving equations involving complex numbers, always express negative square roots in terms of \(i\), and check for both positive and negative solutions!

Use square roots to solve the equation x² = -25 over the complex numbers. Select all solutions for the equation.

Learn how to solve the equation x² = -25 using square roots in complex numbers. This step-by-step guide explains the use of imaginary numbers (i) and provides detailed solutions: x = 5i and x = -5i. Perfect for Grades 9-12.

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