# Solving Math Problems: Trial & Error vs. Other Methods

• FeDeX_LaTeX
In summary, the conversation discusses the concept of using trial and error to solve mathematical equations and whether there are any equations that can only be solved using this method. While there are some equations that can be solved using specific methods, such as the Lambert W function, most equations require numerical methods which are essentially variations of educated trial and error. The conversation also mentions different methods such as the mid-point or bisection method and Newton's method. The conversation ends with an example equation, x + sin(x) = y, to illustrate the concept of using trial and error to find a solution.

#### FeDeX_LaTeX

Gold Member
Hello;

Is there any type of mathematical problem which can only be solved by trial and error (and therefore no other method has been found)? For example, for a cubic equation x^3 + x = 25, one could use trial and error, but a method arrives at the answer too.

Thanks.

Well, exactly what do you mean by a "method"? The equation $$\displaystyle xe^x= 1$$ can be solved by the "Lambert W function", x= W(1), precisely because the Lambert W function is defined as the inverse function to $$\displaystyle f(x)= xe^x$$. But how would you evaluate that? For that matter if found that the solution to a different equation were $$\displaystyle x= e^{\pi}$$, how would you evaluate that? (Using a calculator, of course- and how does the calculator find the value?)

The fact is that almost all equations must be solved by numerical methods and numerical methods are often variations of "educated trial and error". There have been a few threads on this board on the "mid-point" or "bisection" method of solving equations or "Newton's method", which are in essense "trial and error"- you evaluate f(x) and see if it is equal to the value you want. The "educated" part is that you can use how f(x) differs from the desired value to make your next "trial".

Try solving x + sin(x) = y for x.

## What is trial and error method and how does it work?

Trial and error method is a problem-solving strategy that involves trying different solutions until the correct one is found. It works by systematically testing different possibilities and adjusting as needed until the desired outcome is achieved.

## What are the advantages of using trial and error method?

One advantage of trial and error method is that it allows for a hands-on approach to problem-solving. It also encourages creativity and critical thinking as multiple attempts are made to find the solution. Additionally, it can be a useful method for solving complex problems that have multiple variables.

## What are the limitations of relying solely on trial and error method?

The main limitation of trial and error method is that it can be time-consuming and may not always lead to the correct solution. It also does not provide a deeper understanding of the problem and relies heavily on luck and chance. Furthermore, it may not be suitable for more advanced math problems that require a deeper understanding of mathematical concepts.

## What other problem-solving methods can be used in conjunction with trial and error?

Other problem-solving methods that can be used in conjunction with trial and error include algorithms, which are step-by-step procedures for solving a specific type of problem, and heuristics, which are general problem-solving strategies that can be applied to a variety of problems. Additionally, using prior knowledge and logical reasoning can also aid in finding the solution.

## How can one determine when to use trial and error method versus other problem-solving methods?

The decision to use trial and error method versus other methods ultimately depends on the nature of the problem and the individual's strengths and preferences. Trial and error method may be more suitable for problems with multiple solutions or for individuals who learn best through hands-on experimentation. However, for more complex problems, it may be more effective to use other methods that require a deeper understanding and analysis of the problem.