Question

Giải phương trình

\(\left(x-5\right)^4+\left(x-2\right)^4=17\)

Answer 1

Đặt \(x-\frac{7}{2}=a\) phương trình trở thành:

\(\left(a-\frac{3}{2}\right)^4+\left(a-\frac{3}{2}\right)^4=17\)

\(\Leftrightarrow2a^4+12a^2.\left(\frac{3}{2}\right)^2+2\left(\frac{3}{2}\right)^4=17\)

\(\Leftrightarrow2a^4+27a^2-\frac{55}{8}=0\)

\(\Rightarrow\left[{}\begin{matrix}a^2=\frac{1}{4}\\a^2=-\frac{55}{4}< 0\left(l\right)\end{matrix}\right.\)

\(\Rightarrow a=\pm\frac{1}{2}\Rightarrow\left[{}\begin{matrix}x-\frac{7}{2}=\frac{1}{2}\\x-\frac{7}{2}=-\frac{1}{2}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=4\\x=3\end{matrix}\right.\)