Solving A Factorial Equation | How To Solve Factorial Equation Without Using Mathematical Induction?

How to solve a factorial equation x! = x³ - x

[soumyaj]

Nice solution.
I did it using the fact that (x - 2)! > x + 1 for integer x greater than 5 [can be proved by mathematical induction]. So I only have to check for x =1, 2, 3, 4, 5 and x=5 is the solution.

[shhriar]

I wonder if there is any way to solve this other than examining integers one by one in the end. This approach might help with this equation, but it was not helpful if we had to, for example, try 1000 integers to solve the equation. I see we can say x^3-x = x(x^2-1) = x(x+1)(x-1) = (x+1)(x)(x-1) = x! = (x)(x-1)(x-2)!, so eventually (x+1) = (x-2)!, but I could not figure out if it helps with solving the equation via an alternative way.

[palestinemorocco]

Hi, what is the solution if x is complex number ?

[antoniorodriguescajejunior]

Wow! Very good!!!
I want more problem solving.