The equation you provided is:

$x^2 + 1 = 0$

To solve for $x$, follow these steps:

1. Isolate the $x^2$ term: $x^2 = -1$

2. Take the square root of both sides: $x = \pm \sqrt{-1}$

Since $\sqrt{-1}$ is not a real number, we use the imaginary unit $i$ (where $i = \sqrt{-1}$).

Thus, the solutions are:

$x = \pm i$

Therefore, the solutions to the equation are $x = i$ and $x = -i$, which are imaginary numbers.

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Would you like more details on imaginary numbers or their properties? Feel free to ask!

Here are a few related questions to explore further:

1. What is the imaginary unit $i$, and why is it used?
2. How do you solve quadratic equations with complex solutions?
3. What are complex numbers and how are they represented graphically?
4. Can all quadratic equations have real solutions?
5. How do you perform operations on complex numbers?

Tip: Remember, for equations like this, if the square of a real number equals a negative number, the solutions must involve complex or imaginary numbers.