So we have
N + 1/N = 13/6 multiply both sides by N
N2 + 1 = (13/6)N multiply both sides by 6
6N2 + 6 = 13N subtract 13N from both sides
6N2 - 13N + 6 = 0 factor this
(3n -2)(2n -3) = 0
Setting each factor to 0, we have that either N = 2/3 or N = 3/2
Check for yourself that adding either number to its reciprocal gives us 13/6......
