How can I prove sum of the squares of the roots of $x^3+ax^2+bx+c$ is equal to $a^2-2b$?

Question

How would I prove that the sum of the squares of the roots of $x^3+ax^2+bx+c$ is equal to $a^2-2b$?

Answer 1

Say the roots are $r,s,t$. Then $$(x-s)(x-r)(x-t)=x^3+ax^2+bx+c$$

Comparing coefficients we see that $$s+r+t=-a\quad \& \quad sr+st+rt=b$$

It follows that $$a^2=(s+r+t)^2=s^2+r^2+t^2+2(sr+st+rt)$$ and we are done.