What is the method to show exactly one positive root of a Cubic equation? I have $a x^3 + b x^2 + c x + d =0$ and $a>0 , d<0$
$a, b, c, d$ is the function of all parameters.   I'm looking for an analytic solution of this cubic equation. How to prove that is cubic equation has exactly one positive root?

The standard method to define
that the cubic polynomial 
\begin{align} 
f(x)&=ax^3+bx^2+cx+d
\end{align}  
has only one real root
is to check that
its discriminant
$\Delta<0$:
\begin{align} 
\Delta &= 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
,
\end{align}  
Obviously, just the two conditions $a>0$, $d<0$
is not enough to define even if there is only one real root.
