Question

Resolva a equação:
$\frac{x}{1-x} + \frac{x-2}{x}=1$

Answer 1

$\frac{x}{1-x}+\frac{x-2}{x}=1$

M.M.C entre (1 - x) e x: x(1 - x)

Multiplicando a equação por x(1 - x):

$\frac{x}{1-x}+\frac{x-2}{x}=1\\\\\frac{x(1-x)*x}{1-x}+\frac{x(1-x)*(x-2)}{x}=x(1-x)*1\\\\x*x+(1-x)*(x-2)=x*(1-x)\\x^{2}+x-2-x^{2}+2x=x-x^{2}\\x-2+2x=x-x^{2}\\3x-2=x-x^{2}\\x^{2}+3x-x-2=0\\x^{2}+2x-2=0$

$\Delta=b^{2}-4ac\\\Delta=2^{2}-4*1*(-2)\\\Delta=4+8\\\Delta=12\\\Delta=3*4\\\sqrt{\Delta}=\sqrt{3}*\sqrt{4}\\\sqrt{\Delta}=\sqrt{3}*2\\\sqrt{\Delta}=2\sqrt{3}$

$x=\frac{-b\pm\sqrt{\Delta}}{2a}\\\\x=\frac{-2\pm2\sqrt{3}}{2*1}\\\\x=\frac{2*(-1\pm\sqrt{3})}{2}\\x=-1\pm\sqrt{3}$


$x'=-1+\sqrt{3}\\x'=\sqrt{3}-1\\\\x''=-1-\sqrt{3}$

$\boxed{\boxed{S=\{\sqrt{3}-1,~-1-\sqrt{3}\}}}$