PID Controller Subsituition
Okay:

$$\begin{align*}
k_p + \frac{k_i}{s}+k_d\:s&amp;=\frac{k\left(s+z_1\right)\left(s+z_2\right)}{s}\\\\
k_d\:s^2+k_p\:s + k_i&amp;=k\:s^2 + k\left(z_1+z_2\right)s + k\:z_1\:z_2\\\\\therefore\\\\k&amp;=k_d\\\\
k\left(z_1+z_2\right)&amp;=k_p\\\\
k\:z_1\:z_2&amp;=k_i
\end{align*}$$

If you accept that \$k=k_d\$ solves for \$k\$, then that's two remaining equations and two remaining unknowns:

$$\begin{align*}
z_1+z_2&amp;=\frac{k_p}{k_d}\\\\
z_1\cdot z_2&amp;=\frac{k_i}{k_d}
\end{align*}$$

Can you move forward from here using the quadratic equation?