I would like to know whether $q=\langle 3,3,11\rangle$ (a diagonal ternary form) represents $2$ over $\mathbb{Q}_2$ (i.e. whether there exist $x,y,z\in\mathbb{Q}_2^\times$ such that $q(x,y,z)=2$). I have computed the Hasse invariant for $q$; it's $-1$, and I have computed the Hilbert symbol $(-1,-\mathrm{disc}\;q)_2=1$, so $q$ is anisotropic; no help there.
I'm now out of ideas. Anyone know what to do?
First solve $q(x,y,z) = 2$ mod 8, then use Hensel's Lemma to lift to a solution in $\mathbb{Z}_2$.
When working with quadratic forms in characteristic 2, usually it helps to work mod 8. A good reason for this is because a unit in $\mathbb{Z}_2^{\times}$ is a square iff it is $\equiv 1$ (mod 8).