Re: Algebra Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg7649] Re: [mg7624] Algebra Problem*From*: Richard Finley <trfin at fiona.umsmed.edu>*Date*: Tue, 24 Jun 1997 03:36:13 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Robert, You hit the nail on the head for the cause of the problem...(-1)^(1/3) has three solutions only one of which is real and solves your equation, so Mma cannot assume which one you want. If you first load: <<Miscellaneous`RealOnly` and then look at your equation -9 + 9 (-1)^(1/3) + 18 (-1)^(2/3) Mma will now simplify correctly to 0, which shows it is the solution you want. In addition, if you insert (-1)^(1/3) it will simplify it to -1 since it can now ignore the complex solutions. The problem is, if you do your original Solve[ ... ] you still only get the first solution. A better way to proceed is to recognize from the beginning that your equation is simply a quadratic equation (in the variable x^(1/3) ) so there must be 2 solutions, substitute /. x -> u^3, Solve the resulting equation for u (which gives the solutions { u -> -3 }, { u -> 3/2 }, and then simply cube the results to get your x = -27, 27/8. Hope that helps... RF At 04:15 PM 6/20/97 -0400, you wrote: > >I am using Mathematica to study Algebra. When I solve the following >equation from my textbook with Mathematica, it gives me this result: > >In[1]:= >Solve[2x^(2/3) + 3x^(1/3) - 9 == 0] > >Out[1]= > 27 >{{x -> --}} > 8 > >However, my textbook claims that -27 is also a solution. If I tell >Mathematica to turn off its solution verification, it finds the other >textbook solution: > >In[213]:= >Solve[2x^(2/3) + 3x^(1/3) - 9 == 0, VerifySolutions -> False] > >Out[213]= > 27 >{{x -> -27}, {x -> --}} > 8 > >When I ask Mathematica to substitute -27 for x in the equation, >Mathematica only goes so far in simplifying the equation, but not far >enough to determine if the left side is the same as the right side: > >In[2]:= >2x^(2/3) + 3x^(1/3) - 9 == 0 /. x -> -27 > >Out[2]= > 1/3 2/3 >-9 + 9 (-1) + 18 (-1) == 0 > >Now, I can see that the cube root of -1 is -1, and that the taking the >square of -1 yields 1, and then taking the cube root of that also yields >1. So the equation should simplify to: > >-9 - 9 + 18 == 0 > >and then > >0 == 0 > >which would indicate that -27 is indeed a solution. But I can't figure out >how to get Mathematica to return similar results. Trying to force >Mathematica to give a numerical answer, yields an imaginary, non-zero >result: > >In[3]:= >N[2x^(2/3) + 3x^(1/3) - 9 /. x -> -27] > >Out[3]= >-13.5 + 23.3827 I > >which may explain why Mathematica rejects this solution. So, is >Mathematica rejecting the solution -27 appropriately or not? > >Thanks in advance. > >====================================================================== > Robert McNally <mailto:ironwolfNO at SPAMdangerousgames.com> > Visit <http://personalweb.lightside.com/pfiles/mcnally1.html> >---------------------------------------------------------------------- > Finger for my PGP Key -- Protect Your Crypto-Rights! * Free Speech! > Unsolicited Commercial E-Mail Sucks! * Visit http://www.vtw.org >====================================================================== > >