Solving an equation containing cube roots: $\sqrt[3]{5x+7}-\sqrt[3]{5x-12}=1$ I'm trying to figure out a way to solve this equation:
$$\sqrt[3]{5x+7}-\sqrt[3]{5x-12}=1.$$
I tried to cube both sides, but I ended up with an equation looking like this:
$$\sqrt[3]{(5x-12)(5x+7)}(\sqrt[3]{5x-1}-\sqrt[3]{5x+7})=-6.$$
At this point I'm out of stuff to do. Any help would be appreciated. Thanks in advance.

Before cubing, separate your roots:
$$\sqrt[3]{5x+7}=\sqrt[3]{5x-12}+1$$
Now cube:
$$5x + 7 = 5x - 12 + 3\sqrt[3]{5x-12}^2 + 3\sqrt[3]{5x-12} + 1$$
Simplify:
$$ \sqrt[3]{5x-12}^2 + \sqrt[3]{5x-12} - 6 = 0$$
Solve for $\sqrt[3]{5x-12}$ (it is a quadratic equation). Finally solve to $x$.
