Question

Find the condition that the equation $a{x}^3+3b{x}^2+3cx+d=0$ may have two roots equal.

Answer 1

In this case the equations $f\left(x\right)=0$, and $f^{\prime }\left(x\right)=0$, that is
$a{x}^3+3b{x}^2+3cx+d=0....\left(1\right)$,
$a{x}^2+2bx+c=0....\left(2\right)$
must have a common root, and the condition required will be obtained by eliminating $x$ between these two equations.
By combining $\left(1\right)$ and $\left(2\right)$, we have
$b{x}^2+2cx+d=0....\left(3\right)$.
From $\left(2\right)$ and $\left(3\right)$, we obtain
$\frac{x^2}{2(bd-c^2)}=\frac{x}{bc-ad}=\frac{1}{2(ac-b^2)}$;
thus the required condition is
$(bc-ad{)}^2=4(ac-b^2\left)\right(bd-c^2)$.