SOLUTION: if x = (a + (a²+b³)^1/2)^1/3 + (a - (a²+b³)^1/2)^1/3) then x³ + 3bx ise

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Question 1146973: if x = (a + (a²+b³)^1/2)^1/3 + (a - (a²+b³)^1/2)^1/3) then x³ + 3bx ise
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
(a + (a²+b³)^1/2)^(1/3) + (a - (a²+b³)^1/2)^(1/3)
That is equivalent to:



Cardano's cubic formula says that the solution to

x%5E3%2Bpx%2Bq=0 

is 



So if we set

a = -q/2 and b = p/3,

then

q = -2a and p = 3b

then



is the solution to the cubic equation

x%5E3%2B3bx-2a=0

and therefore

x%5E3%2B3bx=2a

answer: 2a

Edwin

Answer by ikleyn(52876) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Looking into the solution by Edwin,  not every school student knows about the formula  Cardano.

            So, I think,  more simple and more accessible solution is expected and is required.

            See such a solution below.  It uses nothing that goes outside of the standard school Math curriculum.

            Nothing --- except combining standard knowledge in the right order :) 


I will introduce new variables  u = %28a%2B%28a%5E2%2Bb%5E3%29%5E%281%2F2%29%29%5E%281%2F3%29  and  v = %28a-%28a%5E2%2Bb%5E3%29%5E%281%2F2%29%29%5E%281%2F3%29.

Then  

    u%5E3 = a%2B%28a%5E2%2Bb%5E3%29%5E%281%2F2%29,

    v%5E3 = a-%28a%5E2%2Bb%5E3%29%5E%281%2F2%29,    

    x = u + v,

    u%5E3%2Av%5E3 = %28a%2B%28a%5E2%2Bb%5E3%29%5E%281%2F2%29%29.%28a-%28a%5E2%2Bb%5E3%29%5E%281%2F2%29%29 = %28a%2Bsqrt%28a%5E2%2Bb%5E3%29%29%2A%28a-sqrt%28a%5E2%2Bb%5E3%29%29 = %28a%5E2+-+%28a%5E2+%2B+b%5E3%29%29 = -b%5E3;

           therefore,  u%2Av = %28-b%5E3%29%5E%281%2F3%29 = -b.


    Thus we have  x = u + v,   u%5E3 + v%5E3 = 2a,  and  u*v = -b.    (1)



Next,  x%5E3 = %28u%2Bv%29%5E3 = u%5E3 + 3u%5E2%2Av + 3u%2Av%5E2 + v%5E3 = u%5E3%2Bv%5E3 + 3uv%2A%28u%2Bv%29.


Substitute here from (1), and you will get


    x%5E3 = 2a + 3*(-b)*x = 2a - 3bx.


Therefore,


    x%5E3 + 3bx = 2a.         ANSWER

Completed and solved,  and an  ELEMENTARY  solution is obtained.