Question 837473
You were correct in setting up the two equations:  n-m=9 and n^2 + m^2 = 261.  Now, you will need to solve the second equation (n^2 + m^2 = 261) by rewriting the first equation (n - m = 9).  You can either solve for n or m.  For this, we will solve for n, which gives us n = 9 + m.  We will now substitute 9 + m for n in equation 2, giving us the following:


(9 + m)^2 + m^2 = 261


We can expand (9 + m)^2.  This will give us:  81 + 18m + m^2.  We now have:


81 + 18m + m^2 + m^2 = 261.  Combining like terms will give us:


81 + 18m + 2m^2 = 261


Next, we will subtract 261 from both sides of the equal sign, which will give us:


-261 + 81 + 18m + 2m^2 = 0


Once again, we will combine like terms, giving us:


-180 + 18m + 2m^2 = 0


I want to simplify this equation to make it easier to solve, so, since -180, 18, and 2 are all divisible by 2, I will divide all of these by 2, giving us:


-90 + 9m + m^2 = 0


Let's rearrange to put this in standard form:  m^2 + 9m - 90 = 0


We can factor this equation, since the factors of -90 that multiply to give us -90 and add together to give us 9 are 15 and -6.  This gives us:


(m + 15)(m - 6) = 0


This gives us:  m = -15 and 6.  Remember, we are looking for integers, and -15 is not an integer.  Therefore, 6 is one of our integers.  To find our other integer, simply replace m in our rewritten equation 1 (n = 9 + m), with 6:


n = 9 + 6, which gives us n = 15.


Therefore our two integers are 6 and 15