Aquestion that had many ways to solve

 Two positive natural numbers are such that the difference of their squares is 11. The sum of these two numbers is:

This question is interesting need method to solve fast and simple an small work.

We see that 11 isn't a big diference so probably is a number under 10 all the number squarred, so the Universe to work is small, ploting a method I gona investigate the number that are next, so quest need planning, you aren't the most trained but you gonna be the most efficient, because you have method, planejamento so here is to see if you are pratical,

2²-1² = 1 very small

9² - 8² = 81 - 64 = 17

6² - 5² = 36 - 25 = 11

If I make some more I gonna see that gonna jump of 2 numbers, type the diference would be 1, 3, 5, 7, 9, 11, 13, 14, so is even passive of formula

y = x² - (x-1)²

After I gonna make sure that isn't possible for other combinaations, but for the question even have other solution the question didn't limited, so the solution is finished I only want to see:

5² - 4² = 9; 25 - 9 = 16, had already blow for 5;

4² - 3² = 16 - 9 = 7; 16 - 4 = 12, already blow for 4;

3² - 2² = 5; 9 - 1 = 8 blow for 3;

But have this consideration is a first degree equation, that qe can put in the geogebra and see that:

So this is the graphic and is wanted where the difference is 11, 

int the end you can change all that conclusion for that x² - (x² - 1) = 11; that is x² - x² + 2x - 1 = 11; and look that laziness 2x - 1 = 11 is equal to x + (x - 1) = 11, but only if you notice that I used like x = x and like y = x -1, what would be wrong? if was x + (x + 1) = 11.

And here seeing why a number that is the neighbour of the neighbour and is because would not be a natural number, but also if have two answer that are correct and is asked only one any one fits Ok.

Rute Bezerra de Menezes Gondim