Suppose I have a cubic equation with unknown coefficient. How can I find the right coefficient to settle the roots condition?

Question

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.

Answer 1

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.

Answer 2

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.