[tex]x+\frac{1}{x}=6;\ D:x\neq0\\\\\frac{x^2}{x}+\frac{1}{x}=6\\\\\frac{x^2+1}{x}=\frac{6}{1}\\\\cross\ multiply\\\\x^2+1=6x\\\\x^2-6x+1=0\\\\x^2-2x(3)+3^2-3^2+1=0[/tex]
[tex](x-3)^2-9+1=0\\\\(x-3)^2-8=0\\\\(x-3)^2=8\iff x-3=\sqrt8\ or\ x-3=-\sqrt8\\\\x=3+\sqrt{4\cdot2}\ or\ x=3-\sqrt{4\cdot2}\\\\\boxed{x=3+2\sqrt2\ or\ x=3-2\sqrt2}[/tex]
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[tex]if:\\\\\frac{x+1}{x}=6;\ D:x\neq0\\\\\frac{x+1}{x}=\frac{6}{1}\\\\cross\ multiply\\\\6x=x+1\\\\6x-x=1\\\\5x=1\ \ \ \ \ |:5\\\\\boxed{x=\frac{1}{5}}[/tex]
