Let the two consecutive positive integers be $x$ and $x + 1$.

The sum of their squares is given as: $x^2 + (x + 1)^2 = 365$

First, expand the equation: $x^2 + (x^2 + 2x + 1) = 365$ $2x^2 + 2x + 1 = 365$

Now, subtract 365 from both sides: $2x^2 + 2x + 1 - 365 = 0$ $2x^2 + 2x - 364 = 0$

Divide the entire equation by 2 to simplify: $x^2 + x - 182 = 0$

Now, solve this quadratic equation using the quadratic formula: [ x = \frac{-b \