k=3n+1/n+2 for n?

Question

formulae

Answer 1

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

Done.

Hope this helps.

:)

Answer 2

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