Question

solve the equation
$x^2+{\left(\frac{x}{x-1}\right)}^2=8$
find the rational root .

Answer 1

$x^2+\frac{x^2}{(x-1{)}^2}=8$
Put $x-1=y\Rightarrow x=y+1$
$\Rightarrow (y+1{)}^2+{(1+\frac{1}{y})}^2=8$
$\Rightarrow y^2+\frac{1}{y^2}+1+2y+\frac{2}{y}+2=8+1=9$
$\Rightarrow {(y+\frac{1}{y}+1)}^2=3^2\Rightarrow y+\frac{1}{y}=\pm 3-1=2,-4$
Hence
$\Rightarrow y^2+1=2y,-4y\Rightarrow y^2+4y+1=0,y^2-2y+1-0$
$\therefore x=1,-2-\sqrt{3},-2+\sqrt{3}$
Hence rational root is $1.$