The equation $x^2 = -1$ has no real solutions, because the square of any real number is always non-negative. However, in the context of complex numbers, there are solutions.

We know that:

$x^2 = -1$

This implies that:

$x = \pm \sqrt{-1}$

The square root of $-1$ is represented by the imaginary unit, denoted as $i$. Thus, the solutions are:

$x = i \quad \text{or} \quad x = -i$

So, the solutions to the equation $x^2 = -1$ are:

$x = \pm i$

Would you like a more detailed explanation or any further help with complex numbers?

5 Related Questions:

1. What is the definition of the imaginary unit $i$?
2. How are complex numbers represented in general?
3. What is the difference between real and complex numbers?
4. How do you add, subtract, multiply, or divide complex numbers?
5. What is the square root of complex numbers and how is it calculated?

Tip: When solving equations involving negative square roots, always consider whether complex numbers might be involved!