Control Systems - How to find dominant poles *without* MATLAB?

[ctrlhelp10]

Homework Statement

Design a lag-lead compensator for the system of Figure 9.37 so that the system will operate with 20% overshoot and a twofold reduction in settling time. Further, the compensated system will exhibit a tenfold improvement in steady-state error for a ramp input.

Homework Equations

$\zeta=\frac{-ln(\frac{percentOS}{100})}{\sqrt(\pi^2+ln^2(\frac{percentOS}{100}))}$

The Attempt at a Solution

Using the equation and 20% overshoot, $\zeta = 0.456$. How do I find the dominant poles by hand, WITHOUT matlab? Every single example in my book and the ones I've tried looking for online ALL use Matlab...

 

[LvW]

Can you solve a quadratic equation?
You have nothing to do than to find the closed-loop function and set the denominator equal to zero. This gives you the pole distribution.

 

[ctrlhelp10]

Can you solve a quadratic equation?
You have nothing to do than to find the closed-loop function and set the denominator equal to zero. This gives you the pole distribution.

Is it simply:
Gol = $\frac{K}{(s)(s+6)(s+10)}$
Gcl = $\frac{Gol}{1+Gol}=\frac{K}{s^3+16s^2+60s+k}$
s^3+16s^2+60s+k = 0

What do I do from here?

 

[magoo]

Leave the denominator in factored form and find the 3 solutions for s(s+6)(s+10) = 0

It should be pretty straightforward.

 

[ctrlhelp10]

Leave the denominator in factored form and find the 3 solutions for s(s+6)(s+10) = 0

It should be pretty straightforward.

The problem asks for a 20% overshoot, and the damping ratio corresponding to that is 0.456. The book says we have to drag the poles in MATLAB until we get our desired damping ratio shown at the bottom, and then the poles are there (-1.79+-3.5j)

Is there a way to do this completely by hand without the use of the Root locus plot or matlab?