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If a > 0 and a - 1/a = 5/6, then find a + 1/a.

a -  1/a =  5/6                 multiply through by a

a^2  - 1  = (5/6)a           multiply through by 6

6a^2  - 6  = 5a              rearrange as  

6a^2  -  5a  -  6  =  0        factor

(3a  + 2) ( 2a - 3)  = 0

Set each factor to 0  and solve for a   and we have that

a = -2/3   (reject)     or     a = 3/2

So

a +  1/a  =

3/2  +  1 / (3/2)  =

3/2 +  2/3  =

13 / 6

(a - 1/a)^2 = a^2 +1/a^2 - 2 = 25/36,

a^2 + 1/a^2 + 2 = 25/36 + 4 = 169/36,

(a + 1/a)^2 = 169/36,

a + 1/a  = 13/6.