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Mathematics Algebra Equation Solving Wolfram Language

How to solve the equation algebraically?

Aleksandr Miller

Posted 2 years ago

Hello, everyone! Can you please help me to solve the equation algebraically: (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/( 2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/( 2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 132 x)/( 2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/( 2 Sqrt[90 + 91 x + 95 x^2]) == 0. I have no idea... Wolfram Mathematica cannot solve it too.

Hello, everyone! Can you please help me to solve the equation algebraically:

(26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/(
2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/(
2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 132 x)/(
2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/(
2 Sqrt[90 + 91 x + 95 x^2]) == 0.

I have no idea... Wolfram Mathematica cannot solve it too.

POSTED BY: Aleksandr Miller

7 Replies

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EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 2 years ago

POSTED BY: EDITORIAL BOARD

Roland Franzius

Posted 2 years ago

POSTED BY: Roland Franzius

Updating Name

Posted 2 years ago

I tried with Maple 2023 ,but 1 hour computation I aborted. Only numerically: eq = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/(2 Sqrt[90 + 91 x + 95 x^2]); NSolve[eq, x, Reals] ({{x -> -0.384779}})

I tried with Maple 2023 ,but 1 hour computation I aborted.

Only numerically:

 eq = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 
       64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 
       74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 
       132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 
       190 x)/(2 Sqrt[90 + 91 x + 95 x^2]);
NSolve[eq, x, Reals]

(*{{x -> -0.384779}}*)

POSTED BY: Updating Name

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 2 years ago

One can get a polynomial in `x` using `GroebnerBasis`. gbx = First[ GroebnerBasis[(26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/(2 Sqrt[90 + 91 x + 95 x^2]), x]]; It is not small. Exponent[gbx, x] Max[Abs[CoefficientList[gbx, x]]] (* Out[610]= 80 Out[611]= \ 1110276819753660237648054091530651383250064170279077722050580451636841\ 585205837760192206271706180210400 ) There are 80 roots. I'll just show the real ones. N[Select[NSolveValues[gbx == 0, x, WorkingPrecision -> 100], Im[#] == 0 &]] ( Out[612]= {-3.81511, -1.94173, -1.39884, -1.36244, -0.926541, \ -0.854442, -0.384779, -0.363983, -0.350656, -0.349146, -0.331095, \ -0.325256, -0.322182, -0.310507, -0.292343, -0.282015, -0.275535, \ -0.265719, -0.222805, -0.146951, 0.0258104, 0.455657} *)

One can get a polynomial in x using GroebnerBasis.

gbx = First[
   GroebnerBasis[(26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 
        64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 
        74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 
        132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 
        190 x)/(2 Sqrt[90 + 91 x + 95 x^2]), x]];

It is not small.

Exponent[gbx, x]
Max[Abs[CoefficientList[gbx, x]]]

(* Out[610]= 80

Out[611]= \
1110276819753660237648054091530651383250064170279077722050580451636841\
585205837760192206271706180210400 *)

There are 80 roots. I'll just show the real ones.

N[Select[NSolveValues[gbx == 0, x, WorkingPrecision -> 100], Im[#] == 0 &]]

(* Out[612]= {-3.81511, -1.94173, -1.39884, -1.36244, -0.926541, \
-0.854442, -0.384779, -0.363983, -0.350656, -0.349146, -0.331095, \
-0.325256, -0.322182, -0.310507, -0.292343, -0.282015, -0.275535, \
-0.265719, -0.222805, -0.146951, 0.0258104, 0.455657} *)

POSTED BY: Daniel Lichtblau

Roland Franzius

Posted 2 years ago

As it seems, none of the values except the known unique one yields a zero. Does anbody know, how to proceed with the GreobnerBasis in case of a set of roots of different polynomials?

POSTED BY: Roland Franzius

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 2 years ago

Roland is quite right, and I missed the step of weeding out parasite solutions. Here is the corrected approach. radexpr = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/(2 Sqrt[49 + 39 xradexpr = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/(2 Sqrt[90 + 91 x + 95 x^2]); gbx = First[GroebnerBasis[radexpr, x]]; Select[NSolve[gbx == 0, x, W + 37 x^2]) + (40 + 132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 190 x)/(2 Sqrt[90 + 91 x + 95 x^2]); gbx = First[GroebnerBasis[radexpr, x]]; Select[NSolve[gbx == 0, x, WorkingPrecision -> 100], Abs[radexpr /. #] < 10^(-20) &] (* Out[622]= {{x -> -0.3847788729136109731409292759642111842393961241324969914766953396637\ 135729983357283916669163862617368}} *)

Roland is quite right, and I missed the step of weeding out parasite solutions. Here is the corrected approach.

radexpr = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 
     64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 
     74 x)/(2 Sqrt[49 + 39 xradexpr = (26 + 18 x)/(2 Sqrt[91 + 26 x + 9 x^2]) + (33 + 
     64 x)/(2 Sqrt[70 + 33 x + 32 x^2]) + (39 + 
     74 x)/(2 Sqrt[49 + 39 x + 37 x^2]) + (40 + 
     132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 
     190 x)/(2 Sqrt[90 + 91 x + 95 x^2]);
gbx = First[GroebnerBasis[radexpr, x]];
Select[NSolve[gbx == 0, x, W + 37 x^2]) + (40 + 
     132 x)/(2 Sqrt[8 + 40 x + 66 x^2]) + (91 + 
     190 x)/(2 Sqrt[90 + 91 x + 95 x^2]);
gbx = First[GroebnerBasis[radexpr, x]];
Select[NSolve[gbx == 0, x, WorkingPrecision -> 100], Abs[radexpr /. #] < 10^(-20) &]

(* Out[622]= {{x ->
-0.3847788729136109731409292759642111842393961241324969914766953396637\
135729983357283916669163862617368}} *)

POSTED BY: Daniel Lichtblau

Debojyoti Mondal

Debojyoti Mondal, Research Scholar at IIEST, INDIA

Posted 10 months ago

Can this be solved approximately? even if I don't get exact solution is there any way to get approximate expression for solution ?

POSTED BY: Debojyoti Mondal

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