Adjoining square root of a discriminant

Question

Let $f(X) = X^2 + bX + c \in K[X]$ be a quadratic polynomial whose discriminant is $\delta^2 = b^2 - 4c$. I understand that if $\alpha_1$ and $\alpha_2$ are the roots of $f$, then $\delta = \alpha_1 - \alpha_2$. But why is it that $K(\delta)$ is a splitting field for $f$? I can see how $K(\alpha_1 - \alpha_2) \subseteq K(\alpha_1, \alpha_2)$, but what justifies the inclusion in the other direction?

Answer 1

The roots of the polynomial can be calculated by completing the square. This gives $$ \alpha_{1,2}=\frac{-b\pm\delta}{2} $$.