Question

Giải phương trình:
\(\sqrt[3]{x+1}+\sqrt[3]{x+2}+\sqrt[3]{x+3}=0\)

Answer 1

\(\sqrt[3]{x+1}+\sqrt[3]{x+2}=\sqrt[3]{-\left(x+3\right)}\\ \Leftrightarrow\left(\sqrt[3]{x+1}\right)^3+\left(\sqrt[3]{x+2}\right)^3+3\sqrt[3]{\left(x+1\right)\left(x+2\right)}\left(\sqrt[3]{x+1}+\sqrt[3]{x+2}\right)=-\left(x+3\right)\\ \Leftrightarrow x+1+x+2-3\sqrt[3]{x^2+3x+2}.\sqrt[3]{x+3}=-x-3\\ \Leftrightarrow3x+6=3\sqrt[3]{x^3+6x^2+11x+6}\\ \Leftrightarrow\left(x+2\right)^3=x^3+6x^2+11x+6\\ \Leftrightarrow x^3+6x^2+12x+8-x^3-6x^2-11x-6=0\Leftrightarrow x=-2\)