$(x-1{)}^4+(x-5{)}^4=82$

Put $y=\frac{(x-1)+(x-5)}{2}=x-3$

x=y+3 The given equation reduces to

$\begin{array}{l}(y+2{)}^4+(y-2{)}^4=82\\ y^4+24y^2-25=0\\ \left(y^2+25\right)\left(y^2-1\right)=0\\ y=5i,\pm 1\\ x-3=5i,\pm 1\\ x=3+5i, 3\pm 1\\ x=3+5i,4,2\end{array}$