\(\sqrt[3]{2x+1}+\sqrt[3]{2x+2}+\sqrt[3]{2x+3}=0\left(1\right)\)
Pt (1) <=> \(\sqrt[3]{2x+1}+\sqrt[3]{2x+2}=-\sqrt[3]{2x+3}\) (*)
\(\Leftrightarrow\left(\sqrt[3]{2x+1}+\sqrt[3]{2x+2}\right)^3=-\left(2x+3\right)\)
\(\Leftrightarrow4x+3+3\sqrt[3]{2x+1}\cdot\sqrt[3]{2x+2}\cdot\left(\sqrt[3]{2x+1}+\sqrt[3]{2x+2}\right)=-\left(2x+3\right)\) (2)
Thay (*) vào (2) ta được:
\(\left(2\right)\Leftrightarrow\sqrt[3]{2x+1}\cdot\sqrt[3]{2x+2}\cdot\sqrt[3]{2x+3}=-2x-2\)
\(\Leftrightarrow\left(2x+1\right)\left(2x+2\right)\left(2x+3\right)=\left(-2x-2\right)^3\)
\(\Leftrightarrow\left(2x+2\right)\cdot\left[\left(2x+2\right)\left(2x+3\right)+\left(2x+2\right)^2\right]=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+2=0\\8x^2+18x+10=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-1\\\left[{}\begin{matrix}x=-1\\x=-\dfrac{5}{4}\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=-\dfrac{5}{4}\end{matrix}\right.\)
Thử lại chỉ có x = -1 thỏa mãn
Vậy pt có 1 nghiệm duy nhất là \(x=-1\)
