Answer
{$\frac{1}{8},8$}
Work Step by Step
First, we add the fractions on the left hand side by taking their LCM. Upon inspection, the LCM is found to be $x$:
$x+\frac{1}{x}=\frac{65}{8}$
$\frac{x(x)+1(1)}{x}=\frac{65}{8}$
$\frac{x^{2}+1}{x}=\frac{65}{8}$
Now, we cross multiply the two fractions in order to create a quardratic equation:
$\frac{x^{2}+1}{x}=\frac{65}{8}$
$8(x^{2}+1)=65(x)$
$8x^{2}+8=65x$
$8x^{2}-65x+8=0$
Now, we use rules of factoring trinomials to solve the equation:
$8x^{2}-65x+8=0$
$8x^{2}-1x-64x+8=0$
$x(8x-1)-8(8x-1)=0$
$(8x-1)(x-8)=0$
$(8x-1)=0$ or $(x-8)=0$
$x=\frac{1}{8}$ or $x=8$
Therefore, the solution is {$\frac{1}{8},8$}.
