How to Solve e^x+e^{3x}-1=0 for x?

[Guero]

how do i solve
$$e^x+e^{3x}-1=0$$
for x?

 

[exequor]

to make it simple let y=e^x

thus giving y^3 + y - 1 = 0
y(y^2 + 1) = 1

i looked at this quickly and i don't think that you would get a solution for x.

 

[HallsofIvy]

Let y= e^x and the equation becomes y³+ y- 1= 0. Can you solve that?

 

[Guero]

how can the answer be x=0?
$$e^{x}+e^{3x}-1=0$$
$$e^{0}+e^{3\times{0}}-1=0$$
$$1+1-1=0$$
$$1=0$$??

 

[exequor]

the answer cannot be zero, i edited my post (sorry about that)

 

[Guero]

no prob, but is there really no solution?

 

[exequor]

well it makes sense to me because this is a cubic function right? how did you go about solving for y?

 

[Guero]

i didn't, i don't know how to

 

[Zurtex]

There is 1 real answer and 2 initial complex answers, being non-simple roots of the cubic equation they all look quite nasty.

 

[Guero]

could someone explain how to find the real answer?

 

[dextercioby]

Check out Mathworld's page on the cubic equation.

Daniel.

 

[Guero]

ok, thanks

 

[closure]

you can set
$$y=e^x$$
then the equation will become $$y+y^3-1=0$$
now you can creat the term $$y^2$$,then minus the same term.