For what integer values of k does sqrt(x)+sqrt(x+1)=k have a rational solution?
2 Answers
Aug 24, 2016
Any positive integer
Explanation:
Given any positive integer
Note that this value of
Then:
sqrt(x) + sqrt(x+1)
=sqrt((k^2-1)^2/(4k^2))+sqrt((k^2-1)^2/(4k^2)+1)
=sqrt((k^2-1)^2/(4k^2))+sqrt(((k^2-1)^2+4k^2)/(4k^2))
=sqrt((k^2-1)^2/(4k^2))+sqrt(((k^2+1)^2)/(4k^2))
=(k^2-1)/(2k)+(k^2+1)/(2k)
=k^2/k
=k
Aug 24, 2016
Explanation:
so
Finally