Question

\[x^3=x+1\]

Answer 1

What is to be done???

Answer 2

find x

Answer 3

no rational roots, one +ve real root and 2 imaginary for sure

Answer 4

x^3=x+1
x(x^2-1)=1
x(x+1)(x-1)=1
Looks like NO SOLUTION.

Answer 5

let x = a siny

we have

a^3 sin^3 y - asiny - 1= 0

mutiply both sides by -4/a^3

=> -4sin^3 y + 4 siny /a^2 + 4/a^3 =0

we have to chose such an "a" that
4/a^2 = 3
=> a =2/sqrt(3) or -2/sqrt(3)

plugging this back,we have

-4 sin^3 y + 3 siny + 4/a^3 =0

sin(3y) = - 4/a^3

3y = sin^-1 ( -4/a^3)
y = (1/3)sin^-1 ( -4/a^3)

we know a so we know y
and thus we know x

hope i did no mistake there..

Answer 6

If exists then 1<x<2 OR -1<x<0

Answer 7

using wolfram alpha,taking a= 2/sqrt(3)
y= -0.52360 + 0.53621 i

thus x = -0.662360 + 0.562277 i approx..

other root should then be conjugate of that..leme confirm that..

Answer 8

its correct !! :D

Answer 9

It's the silver ratio dude.

Answer 10

I read about the silver ratio today. :D

Answer 11

not quite parth, very close

Answer 12

Or was it the plastic number?

Answer 13

I read about the silver ratio and plastic number.

Answer 14

now using sum of roots = 0

we see,,
real root = 2* 0.662360 = 1.324 approx..

Answer 15

hope i was correct throughout ?

Answer 16

I remember that I read it somewhere. Is it the plastic number?

Answer 17

yes parth, but can you express as an exact result please

Answer 18

similar
http://openstudy.com/users/jonask#/updates/50814c79e4b00774b241cc72