<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] $$(\sqrt[3]{x+y} + \sqrt[3]{x-y})^3  = 1$$
$$x+y+3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})\underbrace{(\sqrt[3]{x+y} + \sqrt[3]{x-y})}_1+x-y=1$$
$$2x+3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})=1$$
$$3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})=1-2x$$
$$3(\sqrt[3]{x^2-y^2})=1-2x$$
$$27(x^2-y^2)=(1-2x)^3$$
$$27y^2 = 8 x^3+15 x^2+6 x-1$$
$$27y^2 = (x+1)^2 (8 x-1)$$
$$y = \frac{(x+1) \sqrt{8 x-1}}{3 \sqrt{3}}$$
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