Find all n ? N, such that the equation x-8 n-10 = n x has no solution x.

We get :
x^2 - 8x + 10n - n^2 = 0
For no roots, D<0
64 - 4(10n-n^2) <0
64 - 40n + 4n^2 <0
n^2 - 10n + 16 < 0
(n-8)(n-2) < 0
n = (2,8)
Hence, n can be 3,4,5,6,7 for no solution of x

Thanks
Bharat Bajaj
IIT Delhi
askiitians faculty