Answer
{$\frac{3}{4},\frac{4}{3}$}
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 $x$:
$x+\frac{1}{x}=\frac{25}{12}$
$\frac{x(x)+1(1)}{x}=\frac{25}{12}$
$\frac{x^{2}+1}{x}=\frac{25}{12}$
Now, we cross multiply the two fractions in order to create a quadratic equation:
$\frac{x^{2}+1}{x}=\frac{25}{12}$
$12(x^{2}+1)=25(x)$
$12x^{2}+12=25x$
$12x^{2}-25x+12=0$
Now, we use rules of factoring trinomials to solve the equation:
$12x^{2}-25x+12=0$
$12x^{2}-9x-16x+12=0$
$3x(4x-3)-4(4x-3)=0$
$(4x-3)(3x-4)=0$
$(4x-3)=0$ or $(3x-4)=0$
$x=\frac{3}{4}$ or $x=\frac{4}{3}$
Therefore, the solution is {$\frac{3}{4},\frac{4}{3}$}.
