Finding the X-intercept of a function

[Airp]

Homework Statement

F(x)= x-12x^1/3 Find the X-intercept

Homework Equations

X-intercept means y=0

The Attempt at a Solution

My math teacher gave plus or minus 24sqrt3 as an answer, but I have no idea how he got that! He also got 0, but I do understand that one... I just don't know how to get the 24sqrt3 Could someone help me, please?

 

[Ray Vickson]

Homework Statement

F(x)= x-12x^1/3 Find the X-intercept

Homework Equations

X-intercept means y=0

The Attempt at a Solution

My math teacher gave plus or minus 24sqrt3 as an answer, but I have no idea how he got that! He also got 0, but I do understand that one... I just don't know how to get the 24sqrt3 Could someone help me, please?

Set $u = x^{1/3}$, so your equation becomes $0 = u^3 - 12 u$, or $0 = u(u^2-12)$.

 

[Airp]

Thank You! Quick question: since you put everything to the cube, should you put -12 also to the power three, because that's how I finally got the answer! Thank you so much!

 

[RUber]

The u-substitution is all you need. However, the answer you get is for u. x = u^3. So you will cube your answer.
There should be no reason to cube 12. That would be assuming that
$F(x)^3=(x-12x^{1/3})^3 = x^3 - 12^3 x$ but that is not how powers of polynomials work.
Edit:
However, in this case, since you are assuming that F(x) = 0 (x- intercept), you will get the right solution since:
$0=x-12x^{1/3}$
$x=12x^{1/3}$
$x^3=12^3x$
$0=x^3-12^3x$
There are many ways to get to the answer...just be sure you know why you are doing things.

 

[Airp]

The u-substitution is all you need. However, the answer you get is for u. x = u^3. So you will cube your answer.
There should be no reason to cube 12. That would be assuming that
$F(x)^3=(x-12x^{1/3})^3 = x^3 - 12^3 x$ but that is not how powers of polynomials work.
Edit:
However, in this case, since you are assuming that F(x) = 0 (x- intercept), you will get the right solution since:
$0=x-12x^{1/3}$
$x=12x^{1/3}$
$x^3=12^3x$
$0=x^3-12^3x$
There are many ways to get to the answer...just be sure you know why you are doing things.

Thank you so much! This community really is awesome!