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.

$$a=\sqrt[3]{5x+7},b=\sqrt[3]{5x-12}\\a-b=1\\(a-b)^3=1\\a^3-3a^2b+3ab^2-b^3=1\\a^3-b^3-3ab(a-b)=1\\a^3-b^3-3ab=1\\5x+7-5x+12-3\sqrt[3]{5x+7}\sqrt[3]{5x-12}=1\\-3\sqrt[3]{5x+7}\sqrt[3]{5x-12}=-18\\\sqrt[3]{5x+7}\sqrt[3]{5x-12}=6\\(5x+7)(5x-12)=216\\25x^2-60x+35x-84=216\\25x^2-25x-300=0\\x^2-x-12=0\\x_1=4,x_2=3$$
