I want to find the range of kc that makes s is less than zero

I want to find the range of kc that makes s is less than zero

This is the equation that I have:

16*s^3 + 280*s^2 + 1400*s + 200*Kc + 2000

Thanks...

[Voss]

That expression is not an equation (or an inequality), so any relationship between s and Kc is possible.

Answer 1

[Walter Roberson]

syms s Kc delta real
assume(delta >= 0)
eqn = 16*s^3 + 280*s^2 + 1400*s + 200*Kc + 2000 == delta

eqn = 

sol = solve(eqn, s, 'MaxDegree', 3)

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

sol = 

sol

sol = 

delta_bounds = solve((25*Kc/4 - delta/32 + 625/108)^2 - 5359375/46656 == 0, delta, 'returnconditions', true)

delta_bounds = struct with fields:
         delta: [2×1 sym]
    parameters: [1×0 sym]
    conditions: [2×1 sym]

delta_bounds.delta

ans = 

delta_bounds.conditions

ans = 

You asked that the value of the expression be less than or equal to 0. In the above code, delta represents the amount by which the expression is less than zero. I am going to assume that the quantities involved are not complex-valued.

if delta is positive and too large then (25*Kc/4 - delta/32 + 625/108)^2 will not be large enough to subtract 5359375/46656 from and still get a real-valued quantity out of the sqrt() for solving s. So delta is bounded by that at most, and lower bound zero.

The equation doesn't work out at all if Kc is too small; you would go imaginary.

I don't know, there just might be a configuration of small Kc that lead to the complex conjugate pairs that are the second and third solutions to be real-valued.