Field extension L of degree 2 is a splitting field over F. (Proof)

Question

Since L is a field extension of degree 2 let a,b be its basis over F where a,b are algebraic over F. Then $L=F(a,b)$. By Tower theorem: $2=[F(a,b):F]=[F(a,b):F(a)][F(a):F]$.

If $a \notin F$ then $F(a,b)=F(a)$. Hence, $min_p(a)=(x-a)q(x)$ i.e $deg(q)=1 \rightarrow q(x)=(x-c)$ splits completely in $F(a,b)=F(a)$.

If $a \in F $ implies $B\notin F$ hence $F(a,b)=F(b)$. Proceed as above.

Hence, L is a splitting field.