Prove that if $ac + bc + c^2 < 0$ then equation (usual notation) has two roots We have $a, b, c$ real parameters, $a ≠ 0$.
Prove that $ax^2 + bx + c = 0$ has two different roots ($b^2-4ac > 0$), if $ac + bc + c^2 < 0$

$ac+bc+c^2<0\Rightarrow -ac>bc+c^2$. Now $\Delta=b^2-4ac>b^2+4bc+4c^2=(b+2c)^2>0\Rightarrow$ there are $2$ distinct real roots.
