Solve Equation: 3(e)^(x -1) = 3(x)^2

[Curious6]

Hi, I know how, to obtain the answer to the following equation intuitively but I would like to understand the calculations involved in obtaining the answer, thanks:

3(e)^(x -1) = 3(x)^2

Thanks in advance

 

[VietDao29]

Hi, I know how, to obtain the answer to the following equation intuitively but I would like to understand the calculations involved in obtaining the answer, thanks:

3(e)^(x -1) = 3(x)^2

Thanks in advance

I don't think you can get exact solutions for this equation. First, try plotting it (by some software), then you can use Newton's method to solve the equation.
First, try to change everything to one side, ie:
3e^{x - 1} - 3x² = 0
Let f(x) = 3e^{x - 1} - 3x²
Now choose an x₀ wisely (near the solutions you can see in the graph). Then use:
$$x_{n + 1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$$, then let n increase without bound, your solution will be:
$$x = \lim_{n \rightarrow \infty} x_{n}$$
Can you go from here? :)
By the way, you should note that x = 1 is an obvious solution to this equation.

 

[Curious6]

Okay, thanks :)