Is There an Exact Solution to the Equation \(4^x + 3^x = 12\)?

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Discussion Overview

The discussion revolves around the equation \(4^x + 3^x = 12\) and whether an exact solution exists. Participants explore methods for solving the equation without using computer algebra systems (CAS) and share their experiences and thoughts on the possibility of finding an exact solution.

Discussion Character

  • Exploratory, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants express a desire for an exact solution to the equation, noting that approximate solutions have been found.
  • One participant mentions that they attempted a method similar to another's but only arrived at an approximate solution.
  • Another participant questions the validity of a claim made by someone who suggested they could find an exact solution but did not provide it.
  • One participant reflects on their ability to solve a different equation, \(4^x + 6^x = 9^x\), by transforming it into a homogeneous format, but expresses uncertainty about applying a similar method to the current equation.
  • A participant notes that even using Wolfram Alpha, the results yield numeric approximations rather than an exact solution.
  • A humorous reference is made to a famous quote about a demonstration that cannot be contained, implying skepticism about the existence of an exact solution.

Areas of Agreement / Disagreement

Participants generally agree that they have not found an exact solution to the equation, and multiple competing views remain regarding the possibility of such a solution existing.

Contextual Notes

There are limitations in the discussion regarding the methods available for solving the equation, as well as the potential for typos or misinterpretations of the original problem.

DrLiangMath
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Solve $4^x+3^x=12$ without using CAS?
 
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maxkor said:
View attachment 11883
Nice solution! But if you could find an exact solution it would be much better.
 
MathTutoringByDrLiang said:
Nice solution! But if you could find an exact solution it would be much better.
Do you have an answer?
 
maxkor said:
Do you have an answer?

I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.
 
MathTutoringByDrLiang said:
I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.
I can solve something like $4^x + 6^x = 9^x$ only because I can put it into a homogeneous format. I can't think of any way to do this with $3^x + 4^x = 12$. Even if there were a typo I don't think we can do $3^x + 4^x = 12^x$.

-Dan
 
If you plug it into wolfram alpha, I don't think you can solve this with a CAS either, it's just numeric approximations for you.
 
DrLiangMath said:
I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.

Reminds me of...
Fermat said:
I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.
 
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