SOLUTION: If 9n^2-30n+c is a perfect square for all integers n, what is the value of c?

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Question 842989: If 9n^2-30n+c is a perfect square for all integers n, what is the value of c?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
9n%5E2-30n%2Bc=%283n-k%29%283n-k%29 perfect square
9n%5E2-3kn-3kn%2Bk%5E2
9n%5E2-6kn%2Bk%5E2

Compare the terms and variables in the corresponding positions, especially the terms of n.
30n=6kn
Using very little or no formal analysis, you expect 30=6k meaning that k=5.

CHECK:
%283n-5%29%5E2=9n%5E2-15n-15n%2B25=highlight_green%289n%5E2-30n%2B25%29 Good.
Also, highlight%28c=25%29