According to this calculator:
The possible values for $x$ when $x^2+1=0$ are $i$ and $-i$.
If $$x^2+1=0$$, then why are the possible answers both $i$ and $-i$?
Original equation: $$x^2+1=0$$
subtract $1$ from both side: $$x^2=-1$$
Take the square root of both sides: $$\sqrt{x^2} = \sqrt{-1}$$
Simplify only the left side: $$x=\sqrt{-1}$$
Since $i$ is equal to $\sqrt{-1}$, I replace $\sqrt{-1}$ with $i$: $$x=i$$
Where does the $x=-i$ come from?
On
To answer your questions in order:
...why are the possible answers both $i$ and $-i$?
Because if you replace $x$ with either of them in the equation $x^2+1=0,$ you get a true statement.
Where does the $-i$ come from?
Well, I wouldn't solve that equation like you did. For example, at a point you wrote $\sqrt{ x^2}=\sqrt {-1}.$ But what does this mean, since the operation $\sqrt{}$ is only defined for nonnegative arguments? Rather, I would note that $$x^2+1=x^2-(-1)=x^2-i^2=(x-i)(x+i)=0,$$ etc. However, we don't need all that. The equation $x^2+1=0$ is pretty basic, and we use it to define what we mean by $i,$ by saying that it has the property $i^2=-1.$ Of course it's not the only number with this property.
$x^2-a^2 = (x+a)(x-a)$, so the solutions are $x=a$ and $x=-a$. In other words, the solutions are $x=\pm\sqrt{a^2}$.