There's An Easy Way!

Clearly $x \ne 0,$ so upon dividing through by $x$ the equation becomes

$x - 11 + \dfrac{1}{x} = 0 \Longrightarrow x + \dfrac{1}{x} = \boxed{11}.$

We can see that the product of the roots of the given equation is 1. Which means roots are reciprocal of each other I.e x & 1/x. So what is asked is nothing but the sum of the roots which is clearly 11.

firstly x^2+1=11x or,x+1/x=11

$x(x-11)+1=0 \\ x-11=-\frac{1}{x} \\ -11=-(x+ \frac{1}{x}) \\ x+ \frac{1}{x}= \large \color{#20A900}{\boxed{11}}$

x^2 - 11x +1 =0

x^2 + 1 = 11X hence divide both side by X we get X+1/X=11. Simple.

x^2 - 11x +1 =0

x^2 + 1 = 11x

x+1/x =11 therefore: x+1/x =11

x(x-11+1/x)=0 or,x-11+1/x=0,as x not equal to 0 so,(x+1/x)=11.