"Did you finish the task already?"
Translation:Ты уже сделала задание?
Check out the conjugations in Wiktionary.com... задание is both the nominative and the accusative, and, as you may already know, it is a different word (with a somewhat similar meaning) from задача (accusative = задачу). I'm only a beginner, but I think your sentence is grammatically fine. But задачу would mean you were talking about a "problem" in the sense of a math problem, and not a more generalized "task."
I wrote "Ты уже закончил задачу?" and it was not accepted. What's wrong with this one? KS4711