1. **State the problem:** Solve the equation $x^2 + 1 = 0$ for $x$. 2. **Recall the formula:** To solve quadratic equations of the form $ax^2 + bx + c = 0$, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=0$, and $c=1$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 0^2 - 4 \times 1 \times 1 = -4$$ Since the discriminant is negative, there are no real solutions; the solutions are complex. 4. **Find the complex solutions:** $$x = \frac{-0 \pm \sqrt{-4}}{2 \times 1} = \frac{\pm 2i}{2} = \pm i$$ 5. **Final answer:** The solutions to the equation $x^2 + 1 = 0$ are: $$x = i \quad \text{and} \quad x = -i$$ Here, $i$ is the imaginary unit with the property $i^2 = -1$.