The equation \(x^2 + 1 = 0\) can be solved by isolating \(x\). Here's the step-by-step solution:

1. Start with the given equation:
   \[
   x^2 + 1 = 0
   \]

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

3. Take the square root of both sides. Since the square root of \(-1\) is not a real number, we introduce the imaginary unit \(i\), where \(i^2 = -1\):
   \[
   x = \pm i
   \]

Thus, the solutions are:
\[
x = i \quad \text{or} \quad x = -i
\]

Let me know if you'd like more details or if you have any questions!

Here are 5 related questions to expand your understanding:
1. What is the definition of the imaginary unit \(i\)?
2. How do you solve quadratic equations with real and complex solutions?
3. What is the geometric interpretation of complex numbers on the complex plane?
4. How do you simplify expressions involving powers of \(i\)?
5. What are the applications of imaginary numbers in engineering and physics?

**Tip:** When dealing with negative numbers under the square root, always consider complex solutions using \(i\).

Explore the solution to the quadratic equation x^2 + 1 = 0, involving complex numbers and the imaginary unit i. This problem is perfect for students in Grades 10-12, offering a step-by-step guide to solving quadratic equations with non-real solutions.

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