1. **State the problem:** Solve the equation $x^2 + 1 = 0$ for $x$. 2. **Recall the formula:** To solve $x^2 + 1 = 0$, isolate $x^2$: $$x^2 = -1$$ 3. **Important rule:** The square of a real number cannot be negative. Therefore, there are no real solutions. However, in the complex number system, we use the imaginary unit $i$ where $i^2 = -1$. 4. **Solve for $x$ in complex numbers:** $$x = \pm \sqrt{-1} = \pm i$$ 5. **Final answer:** The solutions to the equation $x^2 + 1 = 0$ are $$x = i \quad \text{and} \quad x = -i$$