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(20064)   (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

 

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



and therefore



answer: 2a

Edwin
Answer by ikleyn(52884)   (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 =   and  v = .

Then  

     = ,

     = ,    

    x = u + v,

     = . =  =  = ;

           therefore,   =  = -b.


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



Next,   =  =  +  +  +  =  + .


Substitute here from (1), and you will get


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


Therefore,


     + 3bx = 2a.         ANSWER

Completed and solved,  and an  ELEMENTARY  solution is obtained.



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