Question

Solve for $x:x=\frac{1}{2-\frac{1}{2-\frac{1}{2-x}}},x\ne 2$

Answer 1

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

$\frac{1}{2-\frac{1}{2-x}}=\frac{2-x}{-2x+3}$

$=\frac{1}{2-\frac{-x+2}{-2x+3}}$

$2-\frac{2-x}{-2x+3}$

$=\frac{2(-2x+3)}{-2x+3}-\frac{2-x}{-2x+3}$

$=\frac{2(-2x+3)-(2-x)}{-2x+3}$

$=\frac{-3x+4}{-2x+3}$

$=\frac{1}{\frac{-3x+4}{-2x+3}}$

$x=\frac{-2x+3}{-3x+4}$

$x(-3x+4)=-2x+3$

$-3x^2+4x=-2x+3$

$-3x^2+4x-3=-2x$

$-3x^2+6x-3=0$

$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-6\pm \sqrt{0}}{2(-3)}$

$x=\frac{-6}{2(-3)}$

$x=1$