Re: Algebra Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg7642] Re: [mg7624] Algebra Problem*From*: seanross at worldnet.att.net*Date*: Tue, 24 Jun 1997 03:36:05 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Robert McNally 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 > ====================================================================== Mathematica has no built-in preference over which root to choose when doing complex operations. Humans, on the other hand, tend to like real numbers. (-27)^(1/3) has three answers as does (-27)^2/3. Two of those roots are complex, one is real. The package 'RealOnly' in the standard add-on packages may help you out. Instead of using the Solve command, you could also check out Roots and FindRoots.