Question

\(\sqrt[3]{x-1}+\sqrt[3]{x-2}=\sqrt[3]{2x-3}\)

Answer 1

Đặt \(\hept{\begin{cases}\sqrt[3]{x-1}=a\\\sqrt[3]{x-2}=b\end{cases}}\)

\(\Rightarrow a^3+b^3=2x-3\)

Từ đó \(PT\Leftrightarrow a+b=\sqrt[3]{a^3+b^3}\)

\(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)=a^3+b^3\)

\(\Leftrightarrow ab\left(a+b\right)=0\)

\(\Leftrightarrow\hept{\begin{cases}a=0\\b=0\\a=-b\end{cases}\Leftrightarrow\hept{\begin{cases}\sqrt[3]{x-1}=0\\\sqrt[3]{x-2}=0\\\sqrt[3]{x-1}=-\sqrt[3]{x-2}\end{cases}}}\)

\(\Leftrightarrow\hept{\begin{cases}x=1\\x=2\\x=1,5\end{cases}}\)