Let the roots of the cubic equation $x^3+ax^2+bx+c=0$ be real.Show that the difference between the greatest and the least of them is not less than $\sqrt{a^2-3b}$ nor greater than $2\sqrt{a^2-3b}$.
Let the real roots of the cubic equation $x^3+ax^2+bx+c=0$ be $\alpha,\beta,\gamma$.Let $\alpha\geq\beta\geq\gamma$.Then: $$\alpha+\beta+\gamma=-a \tag1$$ $$\alpha\beta+\beta\gamma+\gamma\alpha=b \tag2$$ $$\alpha\beta\gamma=-c \tag3$$ I am stuck on how to prove $$\sqrt{a^2-3b}\leq\alpha-\gamma\leq2\sqrt{a^2-3b}$$ Please help me out.Thanks.