Quadratic equations like this one, with an $x^{2}$ term but no $x$ term, can still be solved using the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$, once they are put in standard form: $ax^{2}+bx+c=0$.
$$x^{2}+1=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1$ for $a$, $0$ for $b$, and $1$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
$$x=\frac{0±\sqrt{0^{2}-4}}{2}$$
Square $0$.
$$x=\frac{0±\sqrt{-4}}{2}$$
Take the square root of $-4$.
$$x=\frac{0±2i}{2}$$
Now solve the equation $x=\frac{0±2i}{2}$ when $±$ is plus.
$$x=i$$
Now solve the equation $x=\frac{0±2i}{2}$ when $±$ is minus.
