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The given equation is \( x^2 + 9 = 0 \).

To solve for \( x \):

1. Subtract 9 from both sides:
   \[
   x^2 = -9
   \]

2. Take the square root of both sides:
   \[
   x = \pm \sqrt{-9}
   \]

3. Simplify the square root of a negative number using imaginary numbers:
   \[
   x = \pm 3i
   \]

Thus, the correct solution is \( x = \pm 3i \), which corresponds to the first option.

Would you like further details on solving quadratic equations with complex numbers, or any clarifications?

Here are five related questions for you:
1. What happens when solving an equation with a positive constant, like \( x^2 + 9 = 0 \)?
2. How do you handle square roots of negative numbers in general?
3. What are imaginary numbers, and how do they relate to real numbers?
4. Can a quadratic equation have real and imaginary roots at the same time?
5. What is the significance of the \( i \) (imaginary unit) in complex number theory?

**Tip**: Whenever you encounter a negative under a square root, use \( i \), where \( i = \sqrt{-1} \).

Learn how to solve the quadratic equation x^2 + 9 = 0, which requires using complex numbers. This step-by-step solution involves taking the square root of a negative number and understanding imaginary numbers, suitable for high school algebra students.

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