<s>[INST] Solving implicit equation for x or y I want to solve the following equation for $x$ or $y$ (does not matter wich one) analytically.
$$\sqrt[3]{x+y} + \sqrt[3]{x-y}  = 1$$
Wolframalpha returns following solution, but I could not think of a way how to get there:
$$ x+1 \neq 0 \qquad   y = \frac{(x+1) \sqrt{8 x-1}}{3 \sqrt{3}}$$
Is there a nice 'tool' I do not know for solving this?
 [/INST] Suppose $a+b+c=0$ and that $a,b,c$ are the roots of the equation $$f(x)=x^3-px^2+qx-r=0$$
Then $p=a+b+c=0, r=abc$ and adding $f(a)+f(b)+f(c)=0$ we obtain:$$a^3+b^3+c^3+q(a+b+c)-3r=0$$ whence $$a^3+b^3+c^3=3abc$$
Now let $a=\sqrt[3]{x+y}, b=\sqrt[3]{x-y}, c=-1$ to obtain:$$2x-1=-3\sqrt[3]{x^2-y^2}$$ You can then cube this and isolate $y$ to solve, which should give the answer you are looking for.
Note this method does not make it obvious that $x=-1$ is impossible.
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