Question

Giải phương trình: \(\sqrt[4]{5-x}+\sqrt[4]{x-1}=\sqrt{2}\)

Answer 1

\(\sqrt[4]{5-x}+\sqrt[4]{x-1}=\sqrt{2}\)

Đặt \(\sqrt[4]{5-x}=a;\sqrt[4]{x-1}=m\)

\(\Rightarrow\left\{{}\begin{matrix}a+m=\sqrt{2}\\a^4+m^4=4\end{matrix}\right.\)

\(\Rightarrow a^4+m^4=\left(a+m\right)^4\) \(\left[4=\left(\sqrt{2}\right)^2\right]\)

\(\Leftrightarrow a^4+4a^3m+6a^2m^2+4am^3+m^4=a^4+m^4\)

\(\Leftrightarrow2am\left(2a^2+3am+2m^2\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}a=0\\m=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt[4]{5-x}=0\\\sqrt[4]{x-1}=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=5\\x=1\end{matrix}\right.\) (n)

Vậy phương trình có 2 nghiệm phân biệt . . .