Irreducible components of this variety

Question

Can you give me a clue of how to obtain the irreducible components of $V(y^2 − x^4,x^2 −2x^3 − x^2y + 2xy + y2 − y)$?

Answer 1

First factor $y^2-x^4$ into two parabolas. Then the variety is a subvariety of these two. So the components will be points. (unless the second polynomial contain one of the parabolas which I havent checked)