I think I have the later parts of this proof worked out pretty well but what's really stumping me is how to go from knowing $[K:\mathbb{Q}]=2$ to knowing that $K = \mathbb{Q}[x]/a_2x^2 + a_1x + a_0$.
I mean all I know from $[K:\mathbb{Q}]=2$ is that every element of $K$ can be written in the form $bk_1 + ck_2$ for $b,c\in \mathbb{Q}$. As far as I can tell I don't yet have any theorems at my disposal that say if $[K:\mathbb{Q}]$ is finite than $K$ must be algebraic over $\mathbb{Q}$, or anything like that. How do I go from this premise about $K$ as a 2-dimensional vector space over $\mathbb{Q}$ to knowing something about elements of $K$ as roots of polynomials in $\mathbb{Q}[x]$? Thanks.