Solver misses real solutions



On 10/13/2013 09:35 PM, Aleksas Domarkas wrote:
>  On 13 Oct 2013 06:19:46 Joerg Rauh wrote:
>  > Dear Maxima Supporter,
>  > out of wxMaxima 12.04.0 I had Maxima solve for x:
>  > ((40-3*x)*(20-2*x)*x)/2-500
>  > Here are the results:
>  > 
[x=(-(sqrt(3)*%i)/2-1/2)*((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3)+(1300*((sqrt(3)*%i)/2-1/2))/(81*((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3))+70/9,
>  > 
x=((sqrt(3)*%i)/2-1/2)*((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3)+(1300*(-(sqrt(3)*%i)/2-1/2))/(81*((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3))+70/9,
>  > 
x=((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3)+1300/(81*((250*sqrt(101)*%i)/3^(7/2)+25750/729)^(1/3))
>
>  > It missed the real results: x=1.7407 and x=6.23415, which I found here:
> 
http://www.wolframalpha.com/input/?i=solve%28%28%2840-3*x%29%2F2%29*%2820-2*x%29*x-500%2Cx%29
>
>  > Is there anything I can do differently to make it find the real 
solutions?
>  > Thank you and kind regards
>
>  > Joerg
>
>
>
>  Example. Find real exact solutions of equation:
>  (%i1) eq:((40-3*x)*(20-2*x)*x)/2-500=0 $
>
>  I define function "solvet":
>
>  (%i2) solvet(eq,x):=block([spr,k,maperror,mapprint],
>  /* Copyright (C) 2013 Aleksas Domarkas */
>  maperror:false, mapprint:false,
>  spr:solve(eq,x), rectform(%%),
>  if freeof(sin,%%) then return(sort(%%)) else
>  makelist(x=map(polarform,rhs(spr[k])),k,1,length(spr)),
>  rectform(%%),
>  trigsimp(%%),
>  sort(%%))$
>
>  (%i3) solvet(eq,x);
>  (%o3)
>  [x=(20*sqrt(13)*cos(atan((3^(5/2)*sqrt(101))/103)/3)+70)/9,
>  x=(20*sqrt(13)*cos((atan((3^(5/2)*sqrt(101))/103)-2*%pi)/3)+70)/9,
>  x=(20*sqrt(13)*cos((atan((3^(5/2)*sqrt(101))/103)+2*%pi)/3)+70)/9]
>  (%i4) float(%), numer;
>  (%o4) [x=15.3584874666829,x=6.234149131052292,x=1.740696735598144]
>  (%i5) sort(%);
>  (%o5) [x=1.740696735598144,x=6.234149131052292,x=15.3584874666829]
>
>  (%i6) allroots(eq); /* for test */
>  (%o6) [x=1.740696735598142,x=6.234149131052291,x=15.3584874666829]
>
>  Maple and Mathematica do not solve this problem( find real exact 
solutions).
>

  Nice. I tried with Solve, Reduce, Simplify, FullSimplify.
  FullSimpify and Reduce both just restate that there are three
  roots that solve the equation.

  --John

>
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