I'm familiar with the fact that a smooth quartic $X=Z(F)\subset\Bbb{P}^3$ is a K3 surface. The condition that $K_X\sim 0$ can be shown directly by the adjunction formula, and $H^1(X,\mathcal{O}_X)=0$ follows from a general result (which I'm also familiar with), namely:
If $X\subset \Bbb{P}^n$ is a complete intersection of dimension $d$, then $H^i(X,\mathcal{O}_X)=0$ for all $i=1,...,d-1$.
Now I'm trying to verify that a smooth complete intersection of a quadric and a cubic $Z(F),Z(G)\subset \Bbb{P}^4$ is also a K3 surface.
From the result above, $H^1(X,\mathcal{O}_X)=0$ follows immediately. But how do I compute $K_X$ here?
I imagine the adujnction formula could be useful, by I don't know how.