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

Question

2983 views around the world

You can reuse this answer
Creative Commons License

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

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

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