Yes of course! Though (-i)^2 and i^2 both equal sqrt -1, you are correct that it is from the quadratic equation that you get both of the answers for x.
*EDIT* Actually were both correct. You just did it using the quadratic formula to calculate the answer. I did it from the quadratic equation and resolving as far as x=-1 + sqrt(-1) I just forgot that sqrt(-1) has two answers, -i and +i. What the quadratic formula reminds us of is that there are two answers to any sqrt, the positive and the negative hence ± in the formula.
While I liked math, can't say I miss this stuff. I can say I have never had to do a quadratic equation in the 10 years I spent as a project engineer, then director of engineering. (Or 5 years since.)