Answer:
[tex]\huge\boxed{x=2-2\sqrt2\ \vee\ x=2+2\sqrt2}[/tex]
Step-by-step explanation:
[tex]\dfrac{x-6}{2}=\dfrac{2-x}{x}\\\\\text{Domain:}\ x\neq0\\\\\dfrac{x-6}{2}=\dfrac{2-x}{x}\qquad|\text{cross multiply}\\\\(x-6)(x)=2(2-x)\qquad|\text{use the distributive property}\\\\(x)(x)+(-6)(x)=(2)(2)+(2)(-x)\\\\x^2-6x=4-2x\qquad|\text{add}\ 2x\ \text{to both sides}\\\\x^2-6x+2x=4-2x+2x\\\\x^2-4x=4\qquad|\text{add}\ 2^2\ \text{to both sides}\\\\x^2-2\cdot x\cdot2+2^2=4+2^2\qquad|\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-2)^2=4+4\\\\(x-2)^2=8\iff x-2=\pm\sqrt8\\\\x-2=\pm\sqrt{4\cdot2}[/tex]
[tex]x-2=\pm\sqrt4\cdot\sqrt2\\\\x-2=\pm2\sqrt2\qquad|\text{add 2 to both sides}\\\\x-2+2=-2\sqrt2+2\ \vee\ x-2+2=2\sqrt2+2\\\\\huge\boxed{x=2-2\sqrt2\ \vee\ x=2+2\sqrt2}[/tex]
