Question

Solve the equation $(x-1{)}^4+(x-5{)}^4=82$.

Answer 1

$(x-1{)}^4+(x-5{)}^4=82\Rightarrow \{(x-1{)}^2{\}}^2+\{(x-5{)}^2{\}}^2=82\Rightarrow \{(x-1{)}^2-{\left(x-5\right)}^2{\}}^2+2{\left(x-1\right)}^2{\left(x-5\right)}^2=82\Rightarrow \{x^2-2x+1-x^2+10x-25{\}}^2+2{\left(x^2-2x+1\right)}^2{\left(x^2-10x+25\right)}^2=82\Rightarrow {\left(8x-24\right)}^2+2\left(x^4-10x^3+25x^2-2x^3+20x^2-50x+x^2-10x+25\right)=82\Rightarrow 64\left(x^2-6x+9\right)+2\left(x^4-12x^3+46x^2-60x+25\right)=82\Rightarrow 156x^2-384x+576+2x^4-24x^3+92x^2-120x+50=82\Rightarrow 2x^4-24x^3+248x^2-504x+544=0$