Break in and break away point
This is the characteristic equation
charEq = s^3 + 2 s^2 + 2 s + K == 0;

This is its derivative
dcharEq = D[charEq, s]
(*2 + 4 s + 3 s^2 == 0*)

And if you solve it you get the result you are getting.
sols = NSolve[dcharEq, s]
(* {{s -&gt; -0.666667 - 0.471405 I}, {s -&gt; -0.666667 + 0.471405 I}} *)

Any break-away or break-in point must also satisfy the characteristic equation. If you substitute these solutions in into the characteristic equation you will see it is satisfied for only complex values of \$K\$.
charEq /. sols
(* {(-0.740741 - 0.209513 I) + K == 0, (-0.740741 + 0.209513 I) + K == 0} *)

To correctly formulate the problem you must set the characteristic equation along with its derivative to zero and specify the range of \$K\$.
NSolve[{charEq, dcharEq, K &gt;= 0}, {s, K}]
(* {} *)

The above formulation will give all the points with multiple roots. Te get the break-away or break-in point on the real axis you must also add an additional condition (this is moot in this example).
NSolve[{charEq, dcharEq, K &gt;= 0, s \[Element] Reals}, {s, K}]
(* {} *)