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Re: [Q] Solve?
 To: mathgroup at smc.vnet.net
 Subject: [mg23410] Re: [mg23408] [Q] Solve?
 From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
 Date: Sun, 7 May 2000 21:17:57 0400 (EDT)
 Sender: ownerwrimathgroup at wolfram.com
on 00.5.7 8:14 AM, James at research at proton.csl.uiuc.edu wrote:
>
>
> Hi!
>
> I come across a problem that I don't understand.
> I generate systems of equations
> and try to solve them using mathematica.
> For example,
>
> 
> In[1]= Solve[((4/2)*( 1.50*z+ 2.00)+(44/2)*
> Sqrt[( 1.50*z+ 0.00)^2+ 4.00]) +
> ((4/2)*( 1.50*z+ 2.00)+(44/2)*
> Sqrt[( 1.50*z+ 0.00)^2+ 4.00]) 8.10*z==0, z]
>
> Out[1]= {}
> 
>
> But mathematica can't solve it, simply returing empty.
> This equation can be solved, though, as you see.
> I restarted mathematica to run again, but it doesn't help.
>
>
> The weird thing is some equations are solved, like the below.
> 
> In[2]= Solve[((4/2)*( 1.50*z+ 2.00)+(24/2)*
> Sqrt[( 1.50*z+ 0.00)^2+ 4.00]) +
> ((4/2)*( 1.50*z+ 2.00)+(34/2)*
> Sqrt[( 1.50*z+ 0.00)^2+ 4.00]) 8.10*z==0, z]
>
> Out[2]= {{z>13.4976}}
> 
>
> What's wrong with it and how can I solve it?
> I need to use mathematica, because the number of the systems
> of equations are large.
> (Please disregard any redundunt terms in the equations..
> they are generated by a program.)
> With that, when does mathematica return '{}' as an answer?
> Thank you.
>
>
>  James, newbie in mathematica.
>
>
But Mathematica is completely right: there are no solutions. If you do not
believe it, let's do this "by hand".
After simplification your equation becomes:
In[1]:=
eq = ((4/2)*(1.50*z + 2.00) + (4  4/2)*
Sqrt[(1.50*z + 0.00)^2 + 4.00]) + ((4/2)*(1.50*z + 2.00) + (4

4/2)*Sqrt[(1.50*z + 0.00)^2 + 4.00])  8.10*z == 0 //
FullSimplify
Out[1]=
2
8. + 4 Sqrt[4. + 2.25 z ] == 2.1 z
Let's move the 8 to the RHS:
In[2]:=
% /. Equal[a_ + b_?NumericQ, c_] > Equal[a, c  b]
Out[2]=
2
4 Sqrt[4. + 2.25 z ] == 8. + 2.1 z
Now square both sides and expand:
In[3]:=
Thread[#^2 &[%], Equal] // ExpandAll
Out[3]=
2 2
64. + 36. z == 64.  33.6 z + 4.41 z
Now solve the equation we got:
In[4]:=
Solve[%, z]
Out[4]=
{{z > 1.06363}, {z > 0.}}
Now check if the solutions satisfy the original equation:
In[5]:=
eq /. %
Out[5]=
{False, False}
In other words we only got "parasite" solutions. There are no genuine
solutions at all! If you wanted to see the parasite solutions Mathematica
could have given them in one go:
In[6]:=
Solve[eq, VerifySolutions > False]
Out[6]=
{{z > 1.06363}, {z > 0.}}
Your other equation of course does have a solution and Mathematica correctly
finds it.

Andrzej Kozlowski
Toyama International University
Toyama, Japan
http://sigma.tuins.ac.jp/
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