Find n in terms of K if K=3n+1/n+2

Answer

k=3n+1/n+2
k = ( 3n^2 + 1 +2n ) /n
kn =3n^2 + 1 +2n
3n^2 + 1 +2n -kn =0
3n^2 + (2-k)n + 1 = 0

Just ‘complete the square’. as folows :

3n^2 + (2-k)n + 1 = 0
3 [ n^2 + (2-k)n/3 + 1/3 ] = 0
[ n^2 + (2-k)n/3 + 1/3 ] = 0
[ n + (2-k)/6 ]^2 + 1/3 – (2 -k)^2/36 = 0
[ n + (2-k)/6 ]^2 = (2 -k)^2/36 – 12/36
[ n + (2-k)/6 ]^2 = [ k^2 – 4k – 8 ]/36
[ n + (2-k)/6 ]= SQRT [ k^2 – 4k – 8 ]/6
n = SQRT [ k^2 – 4k – 8 ]/6 – (2-k)/6

Giving,
n ={ SQRT [ k^2 – 4k – 8 ] + k – 2 } / 6