Solving Equation Ʃ(ai^x -b)^2 = C

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Discussion Overview

The discussion revolves around solving the equation Ʃ(ai^x -b)^2 = C for the variable x, where the summation is over a range of indices. Participants explore methods for finding x, including potential analytical and numerical approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant seeks assistance in solving the equation, asking for methods to find x.
  • Another participant suggests expanding the quadratic in the sum to simplify the equation, although there is a clarification that the equation is not a standard quadratic form.
  • A later reply indicates that an analytical solution for the general case may not be possible and recommends numerical methods if specific values for ai are provided.
  • One participant mentions that the problem is part of a research context rather than homework, indicating a different level of complexity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the feasibility of finding an analytical solution, with some suggesting it may not be possible while others focus on numerical approaches. The discussion remains unresolved regarding the best method to solve the equation.

Contextual Notes

There is uncertainty regarding the assumptions needed for solving the equation, particularly in relation to the values of ai and the nature of the summation. The distinction between a quadratic and non-quadratic form is also noted as a point of confusion.

Mikca10
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Hi,

I want to solve the following equation by x:



Ʃ(ai^x -b)^2 = C ,
where Ʃ is over all i, i = 1:N, and ^ means "to the power of"

How to find x from here?

Thanks!
 
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Mikca10 said:
Hi,

I want to solve the following equation by x:



Ʃ(ai^x -b)^2 = C ,
where Ʃ is over all i, i = 1:N, and ^ means "to the power of"

How to find x from here?

Thanks!

Welcome to the PF.

What is the context of the question? Is it from schoolwork?
 
Try expanding the quadratic that's in the sum. Then you should ultimately be able to turn it into one quadratic equation.
 
alexfloo said:
Try expanding the quadratic that's in the sum.

It isn't a quadratic. The equation is [itex]\sum(a_i^x -b)^2 = C[/itex] , not [itex]\sum(a_ix -b)^2 = C[/itex]
 
Ah my mistake. Thanks a lot.
 
it is not the homework, it's more like a part of a research problem.. my algebra class was long time ago.
 
Anyone has a clue?
 
If you want an analytical solution for the general case, I'd say (pretty confidently) that it just can't be done. If you have specific values for the a_i, then I'd try it numerically.
 

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