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Re: Algebra Problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg7676] Re: [mg7624] Algebra Problem
*From*: Allan Hayes <hay at haystack.demon.co.uk>
*Date*: Thu, 26 Jun 1997 01:36:59 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
>From: ironwolfNO at SPAMdangerousgames.com (Robert McNally)
>Subject: [mg7676] [mg7624] Algebra Problem
Robert,
For Mma, (-1)(1/3) is
ComplexExpand[(-1)^(1/3)]
1/2 + (I Sqrt[3])/2
r2 =ComplexExpand[(-1)^(2/3)]
-(1/2) + (I Sqrt[3])/2
Which gives
1/3 2/3
-9 + 9 (-1) + 18 (-1)
as
-9+9r1 +18 r2//Expand
-(27/2) + (27 I Sqrt[3])/2
You can use an equivelent pair of equations - inlude the definition
of the cube roots:
Solve[{2u^2 + 3u - 9 == 0, u^3==x},x]
{{x -> -27}, {x -> 27/8}}
Allan Hayes
hay at haystack.demon.co.uk
http://www.haystack.demon.co.uk/training.html
voice:+44 (0)116 2714198
fax: +44 (0)116 2718642
Leicester, UK
****
Begin forwarded message:
>From: ironwolfNO at SPAMdangerousgames.com (Robert McNally)
>Subject: [mg7676] [mg7624] Algebra Problem
>Organization: Somewhere Out There
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>
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