Question 697461
Call the smaller of the two integers x. The larger integer is one higher, so it equals x+1. We are told the sum of their squares is 145, so:<br>

{{{x^2 + (x+1)^2 = 145}}}, simplifying yields:
{{{x^2 + x^2 + 2x + 1 = 145}}}, giving a quadratic equation. Set one side equal to 0:
{{{2*x^2 + 2x + 1 = 145}}}
{{{2*x^2 + 2x -144 = 0}}}, simplfy by dividing both sides by 2:
{{{x^2 + x - 72 = 0}}}, which can be factored as<br>

(x-8)(x+9) = 0, so x = 8 or -9. Since you are looking for positive integers, x = 8 is the smaller number, and the larger number is 8 + 1 = 9. A quick check:<br>

{{{8^2 + 9^2 = 64 + 81 = 145}}}.