Find two consecutive positive integers such that the sum of their squares is 85.
n^2+(n+1)^2+2n = 85
n^2+n^2+2n+1=85
2n^2+2n=84
n^2+n=42
n^2+n-42=0
(n-6)(n+7)=0
n=6
n=-7
Is my work and answer correct?
-7 is not a positive integer. Your first equation is wrong, because the 2n term should not be there. Your second equation is correct, and after that you correectly solve for n. Since they ask for postive integer solutions only, the only number that fits, for n, is 6. The second integer will by definition be n+1, which is 7.
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