Question 68069
Let x be the first positive integer.  The next consecutive positive integer would then be x+1.
The sum of the squares of these is 85.
{{{x^2 + (x+1)^2 = 85}}} Simplify and solve for x.
{{{x^2 + (x^2 + 2x + 1) = 85}}} Simplify.
{{{2x^2 + 2x + 1 = 85}}} Subtract 85 from both sides.
{{{2x^2 + 2x - 84 = 0}}} Divide through by 2.
{{{x^1 + x - 42 = 0}}} Factor.
{{{(x+7)(x-6) = 0}}} Apply the zero product principle.
{{{x+7 = 0}}} and/or {{{x-6 = 0}}}
So, x = -7 and x = 6  Discard the negative value as the problem requires positive integers.
The two integers are: 6 and 7

Check:
{{{6^2 + 7^2 = 36 + 49}}} = {{{85}}}