Ta có: \(\left(x+1\right)^4+\left(3-x\right)^4=82\)
\(\Leftrightarrow\left(x+1\right)^4+\left(x-3\right)^4=82\) (1)
Đặt \(y=x-1\)
Khi đó: \(\left(1\right)\Leftrightarrow\left(y+2\right)^4+\left(y-2\right)^4=82\)
\(\Leftrightarrow\left(y^4+8y^3+24y^2+32y+16\right)+\left(y^4-8y^3+24y^2-32y+16\right)=82\)
\(\Leftrightarrow2y^4+48y^2-50=0\)
\(\Leftrightarrow y^4+24y^2-25=0\)
\(\Leftrightarrow y^4+25y^2-y^2-25=0\)
\(\Leftrightarrow y^2\left(y^2-1\right)+25\left(y^2-1\right)=0\)
\(\Leftrightarrow\left(y^2+25\right)\left(y^2-1\right)=0\)
\(\Leftrightarrow y=\pm1\)
\(\Rightarrow x=2;x=0\)
Vậy \(x\in\left\{0;2\right\}\)
