Solve for #x# in #root(3)(3+sqrt(x)) +root(3)(3-sqrt(x))= c# ?

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Answer 1 · 2017-02-21T18:36:47

See below.

Supposing that the proposition is

#root(3)(3+sqrt(x)) +root(3)(3-sqrt(x))= c#

Making #z^3=3+sqrtx# we have

#c-z=root(3)(6-z^3)# Cubing both sides

#(c-z)^3=6-z^3# or

#c^3-3c^2z+3c z^2 -6 = 0# solving for #z#

#z = (3 c^2pm sqrt[3] sqrt[12 c - c^4])/(6 c)#

and following

#3+sqrtx=( (3 c^2pm sqrt[3] sqrt[24 c - c^4])/(6 c))^3#

with the final outcome

#x = (( (3 c^2pm sqrt[3] sqrt[42 c - c^4])/(6 c))^3-3)^2#