suppose F is an extension field of K of degree 2. this means that F is a K-vector space of dimension 2. so we can find a u in F with {1,u} as a basis. this means that

u^2 (which is in F), can be expressed as u^2 = au + b, for some a,b in K. so u is a root of x^2 - ax - b in K[x], and -u is another root, meaning K is a splitting field

of x^2 - ax - b.