<s>[INST] 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.
 [/INST] Put $a = \sqrt[3]{5x+7}, b = \sqrt[3]{5x-12}\implies a - b = 1, a^3 - b^3 =19\implies a^2+ab+b^2 = \dfrac{a^3-b^3}{a-b} = \dfrac{19}{1} = 19\implies (a-b)^2+3ab = 19\implies 1^2+3ab = 19 \implies ab = 6\implies (b+1)b = 6\implies b^2+b-6 = 0\implies (b+3)(b-2)=0\implies b = 2, -3\implies 5x-12 = 2^3, (-3)^3 = 8,-27\implies 5x = 20, -15\implies x = 4, -3.$
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