x+(1/x)=5

x + 1/x   = 5        multiply through by x

x^2  + 1  =  5x    subtract 5x from both sides

x^2  - 5x  +  1   = 0      subtract 1 from both sides

x^2 -5x  =  -1          take 1/2 of 5  = 5/2, square it  = 25/4  and add this to both sides

x^2  - 5x  +  25/4   =   -1 +  25/4       factor the left side, simplify the right

(x - 5/2)^2   =  21/4        take pos/neg roots 

x -  5/2   =  ± √ (21/4)  add 5/2 to both sides, simplify the right

x  =    ± √(21) / 2 +  5/2

x  = [  5 ± √21 ]  /  2