Some questions about Milne's algebraic geometry notes. I'm trying to go through the details of an example in J. Milne's algebraic geometry notes (p.42, 2.22). 
He gives us the general fact that for two algebraic subsets $W,W'\subseteq K^n$, we have $I(W\cap W')=rad(I(W)+I(W'))$$\;^{(1)}$, which I believe I understand, but then he goes on to illustrate this with a concrete example.
In his example, I presume we are working in $\mathbb{C}[x,y]$. He sets $W=V(x^2-y)$ and $W'=(x^2+y)$, and then claims that $$I(W\cap W')=rad(x^2, y)=(x,y)\,.\;^{(2)}$$ This is what I would like to understand.
First, I have shown that $(x^2-y)$ and $(x^2+y)$ are prime ideals, and thus radical. This means that $I(W)=IV(x^2-y)=(x^2-y)$ and similarly $I(W')=IV(x^2+y)=(x^2+y)$. So if we believe (1), then we get $$I(W\cap W')=rad\bigl((x^2-y)+(x^2+y)\bigr)\,.$$
So, the first equality of (2) suggest that $(x^2-y)+(x^2+y)=(x^2, y)$. Is this because I can write $x^2$ and $y$ as linear combinations of the generators $x^2-y$ and $x^2+y$? Should this be "obvious"?
Also, since $(x,y)$ is a maximal ideal it is radical, so we have $(x,y)=rad(x,y)$, and thus the second equality of (2) can be written as $rad(x,y)=rad(x^2,y)$, and this equality I would also like to understand. I have convinced myself that $rad(x^2,y)\subseteq rad(x,y)$, but I'm having trouble with the other containment.
Thank you for your help!

It "should" be obvious that $(x^2-y)+(x^2+y)=(x^2,y)$ because the left hand side is equal to $(x^2-y,x^2+y)$ by definition, and two ideals whose generators are contained in each other must be equal. But if it's not obvious, then prove the following carefully as an exercise: if $I=(i_1,...,i_n)$ and $J=(j_1,...,j_m)$ are two ideals (of any commutative ring, or left ideals of an arbitrary ring, or (left) modules over any ring, or...) such that $i_1,...,i_n\in J$ and $j_1,...,j_m\in I$ then $I=J$. The radical of $(x^2,y)$ contains $(x,y)$ because a power of $x$ is in $(x^2,y)$, and since $(x,y)$ is maximal, the radical of $(x^2,y)$ can't be any bigger! Note that none of this needed us to work over $\mathbb{C}$: any field not of characteristic $2$ will do.
