# Trial and Error

Gold Member

## Main Question or Discussion Point

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.

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