Use quadratic formula

or

x^2 = -2i

Let x = a + bi (a,b real numbers)

(a + bi)^2 = -2i

a^2 - b^2 + 2abi = -2i

2ab = -2 gives ab = -1 and a^2 - b^2 = 0

a = -1/b so (-1/b)^2 - b^2 = 0

1/b^2 - b^2 = 0

Multiply by b^2

1 - b^4 = 0

b^4 = 1 gives b = 1 or b = -1

b = 1 gives a = -1/1 = -1 giving solution x = -1 + i

b = -1 gives a = -1/-1 = 1 giving solution x = 1 - i

This methos should be used if you haven't done De Moivre's theorem