let x^2 = w then x^4 = w^2
It is true that if ab= 0, then either a= 0 or b= 0 but that is a property of 0 only!
Saying that does NOT imply that " " or that " ".
As dwsmith said, you original equation is a quadratic in . By the way, I would be inclined to write those last two roots as (1/2)i and (-1/2)i or even -i/2, i/2. -1/2i and 1/2i are too likely to be interpreted as -1/(2i) and 1/(2i) (which, it suddenly occurs to me, are exactly (1/2)i and (-1/2)i!)