To solve the quadratic equation $x^2 + 1 = 0$, we need to find the values of $x$ that satisfy the equation.

1. Start by isolating $x^2$: $x^2 = -1$

2. To solve for $x$, take the square root of both sides. Remember that taking the square root of a negative number involves imaginary numbers: $x = \pm \sqrt{-1}$

3. By definition, $\sqrt{-1} = i$ (where $i$ is the imaginary unit): $x = \pm i$

Therefore, the solutions to the equation $x^2 + 1 = 0$ are: $x = i \quad \text{and} \quad x = -i$

Would you like further details or have any questions?

Here are some questions you might consider next:

1. What are complex numbers and how are they represented?
2. How do you add, subtract, multiply, and divide complex numbers?
3. What are the properties of the imaginary unit $i$?
4. How do you graph complex numbers on the complex plane?
5. What are some applications of complex numbers in engineering and physics?

Tip: When solving quadratic equations, if the discriminant is negative, the solutions will involve imaginary numbers, indicating that the roots are complex.