1. x-intercepts with cubic

Hi all,

Looking for some help to find if there is a "proper" way for doing this which I'm sure there is.

This is the equation.

$x^3 - 3x + 5$

I'm doing lots of things to it like finding turning points etc etc, but I need to sketch it and show important points.

To do this, I need to find the x-intercept.

I really just don't know how to factorize this and get the correct answer. I know (via plugging into graphing calculator) that there is only one x intercept, and it's -2.27902. However I wish to do this 'properly' and learn how to find the factor on my own.

Factorizing doesn't seem to help, because the usual solution I would do is this.

$x^3 - 3x = -5$
$x^2 (x - 3) = -5$
$x = +/- \sqrt{-5}$ (impossible for my purposes) or $x = -2$

Any help as to the proper way to do this without trial and error would be appreciated.

Cheers.

2. Originally Posted by Peleus
$x^3 - 3x = -5$
$x^2 (x - 3) = -5$
Your second line is wrong.. It is

$x^3 - 3x = -5$
$x (x^2 - 3) = -5$

3. You need to rearrange equation to get x = cube root of (3x - 5)

Then you use the iteration formula x(n+1) = cube root(3x(n) - 5)

and it gives you x = -2.279018786 after a number of iterations

4. Originally Posted by SENTINEL4
Your second line is wrong.. It is

$x^3 - 3x = -5$
$x (x^2 - 3) = -5$
Woops, my silly mistake, I suppose the point still stands though because -5 and +/- root -2 aren't solutions.

5. Originally Posted by mathswend
You need to rearrange equation to get x = cube root of (3x - 5)

Then you use the iteration formula x(n+1) = cube root(3x(n) - 5)

and it gives you x = -2.279018786 after a number of iterations
Cool, as far as my memory goes I've never heard of / seen the iteration formula, can you point me in the right direction for learning this? Simply google or if you can briefly explain it would be awesome.

Edit - Or is it simply another name for trial and error?

6. you simply rearrange your formula to get x =....

in this case x = cube root(3x - 5)

then you choose a starting point for x... say x = 1

second value is cube root(3(1) - 5) = -1.25992105
third value is cube root(3(-1.25992105) - 5) = -2.062976433
fourth value is cube root(3(-2.062976433) - 5) = -2.236640458

this keeps getting closer and closer to the root of the equation, so the more you do it the better your answer.

Hope this helps.