solving a cubic equation

I need to find the minimum $r$ and the maximum $k$ of the following cubic equation for which there does not exist three distinct real roots.

$rx^3-rkx^2+(r+k)x-rk=0$.

Is it possible to find such $r$ and $k$ analytically? Or if you can provide me help using mathematica, that would be fine too.Thanks.

• You could try interpreting the output of Reduce[Discriminant[r x^3 - r k x^2 + (r + k) x - r k, x] == 0, {r, k}, Reals] // FullSimplify. May 5, 2013 at 18:19
• Maybe you could tell us the approach(es) you've tried? May 5, 2013 at 18:20
• The result looks really messy. I could not interpret it. Can you give me the solution? May 5, 2013 at 18:27
• I upvoted your question, and then saw your comment "The result looks really messy. I could not interpret it. Can you give me the solution?" Please be aware that if you can't interpret your own problem results, the problem is (perhaps) above your abilities May 5, 2013 at 19:05
• You want the cubic polynomial's discriminant to be non-positive for it not to have 3 distinct real roots. If you look at the results of what @J.M. suggested, you should at least be able to handle readily some of the boundary cases that arise. May 5, 2013 at 19:59

Solve[r*x^3 - r*k*x^2 + (r + k)*x - r*k == 0, x]

• I agree that it should probably be a comment but I couldn't help upvoting, as I think it's an interesting point. For specific irreducible cubics, you can often (always??) express the roots in this form using ComplexExpand. For example: ComplexExpand[Re[z /. Solve[z^3 - 3 z - 1 == 0, z]]]. In this case, the Re simply removes the imaginary parts, which we know to be zero anyway. May 6, 2013 at 13:06