What are the root of this polynomial which has no real roots

Question

One of the exercises in control systems asks to find the roots of this characteristic equation if $K=1$ $$ (1-K)s^2 + 2(1-K)s + (2-K) = 0 $$ Obviously it has no real roots but is it possible to extend this to the complex plane and if so how one can compute it. Notice: in control systems we use complex plane a lot. This is why I'm thinking there is a way to solve it but I don't know how.

Answer 1

Your equation can also be written

$$ (1 - K)(s^2 + 2s + 1) + 1 = 0, $$

which is equivalent to

$$ 1 = (K - 1)(s+1)^2.$$

As $K$ goes to $1,$ then $K - 1$ goes to $0$ and $\lvert s\rvert$ goes to infinity. The direction from which $K - 1$ approaches $1$ determines which directions $s$ can go; for any value of $K$ other than $1$ there are two choices of $s$.