Question

Giải phương trình: \((x-5)^4+(x-2)^4=17\)

Answer 1

Lời giải:

Đặt \(x-\frac{7}{2}=a\). Khi đó PT trở thành:

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

\(\Leftrightarrow 2a^4+27a^2+\frac{81}{8}=17\)

\(\Leftrightarrow 2a^4+27a^2=\frac{55}{8}\)

\(\Leftrightarrow a^4+\frac{27}{2}a^2=\frac{55}{16}\)

\(\Leftrightarrow (a^2+\frac{27}{4})^2=49\)

\(\Rightarrow \left[\begin{matrix} a^2+\frac{27}{4}=7\\ a^2+\frac{27}{4}=-7< 0(\text{vô lý})\end{matrix}\right.\)

\(\Rightarrow a^2=\frac{1}{4}\Rightarrow a=\pm \frac{1}{2}\)

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