Question

Find the sum of the roots of the equation $2^{333x-2}+2^{111x+1}=2^{222x+2}+1$

Answer 1

let $2^{111x}=y$

$\therefore$ equation becomes

$y^3-2^{-2}+2-y=2^2-y^2+1$

$\therefore y^3+8y=16y^2+1$

$y^3-16y^2+8y-1=0$

$y_1{y}_2{y}_3=2^{111(x_1+x_2+x_3)}$

$\Rightarrow x_1+x_2+x_3=\frac{\log \left(y_1{y}_2{y}_3\right)}{111\log \left(2\right)}$

$y_1{y}_2{y}_3=1$

$\therefore x_1+x_2+x_3=\frac{{\log }^1}{111{\log }^2}$

$=0$