I'm using the book "Control Systems Engineering - Norman S. Nise" and in the Root Locus chapter most of the exersices are solved using software.
I'm aware that this is a practical solution to real problems, however I'm curious how can I get a closed form solution to problems of the type :
Find a value of K for which the poles of the negative feedback closed-loop system would have a $\zeta = 0.707 $
Assume the open loop transfer function is
$$ G(s) = \frac{K(s-1)(s-2)}{(s+1)(s+2)} $$ $$ H(s) = 1 $$
The closed-loop transfer function is, \begin{align} y &= G(x-z) \\ &= G(x-Hy) \\ (1+GH)y &= Gx \\ y &= \frac{G}{1+GH}x \end{align} Let $n_G$, $n_H$ be the numerator of $G$ and $H$, and $d_G$, $d_H$ be the denominator of $G$ and $H$.
\begin{align} y &= \frac{G}{1+GH}x \\ &= \frac{\frac{n_G}{d_G}}{1+\frac{n_G}{d_G}\frac{n_H}{d_H}}x \\ &= \frac{n_Gd_H}{n_Gn_H + d_Gd_H}x \\ &= \frac{K(s-1)(s-2)}{K(s-1)(s-2) + (s+1)(s+2)}x \\ &= \frac{K(s-1)(s-2)}{K(s^2-3s+2) + (s^2+3s+2)}x \\ &= \frac{K(s-1)(s-2)}{(1+K)\left[s^2+\frac{3(1-K)}{1+K}s+\frac{2+K}{1+K}\right]}x \end{align}
$\omega_n^2 = \frac{2+K}{1+K}$ and $2\zeta\omega_n = \frac{3(1-K)}{1+K}$. Therefore, \begin{align} \zeta &= \frac{3(1-K)}{2\omega_n} = \frac{3(1-K)}{2\sqrt{2+K}\sqrt{1+K}} \end{align}
If you want $\zeta = \frac{1}{\sqrt{2}}$ then $K= \frac{12-\sqrt{109}}{7} $