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Suppose I have this equation, $S^3 + 19S^2 + 25S - 75 + K=0$

How can I find $K$ that makes all the root values negative?

So far, all I'm doing is substituting $K$ with random values until it gives me negative roots. But this trial and error method is time consuming. There has to be a smarter way!

EDIT That $K$ should be the minimum positive value.

Abdulrahman
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2 Answers2

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It wont work. Take a look at the derivative and you will see that the derivative is always positive. You will only get one zero for any K.

Update... consider $y = s^3 + 19s^2 + 25s - 75$

Take the derivative. Find the zeros. These will give you the values of s where y makes a local extrema.

Find y for each of these values of s. That will give you your bounds for K. On futher inspection... this will give you the largest K for which there are 3 roots (of any sign). The smallest K is 75.

Or more precisely K>75. However, there is no "smallest number" greater than 75.

Doug M
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  • He changed the values – imranfat Mar 23 '16 at 21:03
  • I managed to find $K$ by luck. It is 100. But I don't know if it's the smallest +ve value. I'll try your method and see what will happen

    EDIT: I took the derivative and the roots are $S1$ = -0.7 & $S2$ = -11.97

     – Abdulrahman Mar 23 '16 at 21:11
  • @Abdulrahman For $K=100$, there are two complex roots, so they fail to be all negative. – Jean-Claude Arbaut Mar 23 '16 at 21:29
  • @Jean-ClaudeArbaut How so? I managed to get two negative complex roots and one negative real value. This satisfied the condition. But it's not the smallest number. – Abdulrahman Mar 23 '16 at 21:38
  • @Abdulrahman Please define a "negative complex number". – Jean-Claude Arbaut Mar 23 '16 at 21:41
  • @Jean-ClaudeArbaut Well, as far as our instructor said, he solved similar question and ended up with complex roots, too. This is not just any normal equation. This is a characteristic equation that is used to stabilize an unstable system, so all the poles should be in the negative side for the system to be stable. – Abdulrahman Mar 23 '16 at 21:44
  • @Abdulrahman "Negative side" means complex numbers with negative real part. Not the same as negative numbers, which not only have negative real part, but also zero imaginary part. You should clarify this in the question. – Jean-Claude Arbaut Mar 23 '16 at 21:47
  • @Jean-ClaudeArbaut Yes, I meant the real number part of the root. – Abdulrahman Mar 23 '16 at 21:51
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Split the graph into monotonic sections by solving the derivatives.

Find the minimum y shift required to make each of those completely positive or shift the positive root just beyond the 0.

Choose the maximum of the 3 shifts.

user2277550
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