# 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.

## Actually, your work and answer are almost correct, but there was a small mistake in your first equation. The correct equation should be:

n^2 + (n+1)^2 = 85

Let's go through the solution step by step:

1. Write down the equation: n^2 + (n+1)^2 = 85

2. Expand the equation: n^2 + (n+1)(n+1) = 85

n^2 + (n^2 + 2n + 1) = 85

2n^2 + 2n + 1 = 85

3. Rearrange the equation: 2n^2 + 2n - 84 = 0

4. Simplify the equation by dividing all terms by 2: n^2 + n - 42 = 0

5. Factor the equation: (n - 6)(n + 7) = 0

6. Set each factor equal to zero: n - 6 = 0 or n + 7 = 0

n = 6 or n = -7

Since we are looking for positive integers, we can disregard the solution n = -7.

Therefore, the consecutive positive integers that satisfy the equation are n = 6 and n+1 = 7.