SOLUTION: the difference of two integers is 9. The sum of their square is 261. find the two integers n-m=9 n^2 + m^2 = 261 n^2 + 2mn + m^2 = 261 that is all I got

Question 837473: the difference of two integers is 9. The sum of their square is 261. find the two integers
n-m=9
n^2 + m^2 = 261
n^2 + 2mn + m^2 = 261
that is all I got

Found 3 solutions by Alan3354, lwsshak3, josh_jordan:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
the difference of two integers is 9. The sum of their square is 261. find the two integers
n-m=9 --> n = m+9
n^2 + m^2 = 261
n^2 + 2mn + m^2 = 261 *****
-----
n-m=9 --> n = m+9
n^2 + m^2 = 261 Sub for n
(m+9)^2 + m^2 = 261
2m^2 + 18m + 81 = 261

m = -15, m = 6
that is all I got

Answer by lwsshak3(11628)   (Show Source): You can put this solution on YOUR website!
the difference of two integers is 9. The sum of their square is 261. find the two integers
***
n-m=9
n=m+9
n^2+m^2=261
sub n
(m+9)^2+m^2=261
m^2+18m+m^2=261
2m^2+18m=180
2m^2+18m-180=0
m^2+9m-90=0
(m-6)(m+15)=0
m=-15(reject, m > 0)
m=6
n=m+9=15
the two integers: 6 and 15

Answer by josh_jordan(263)   (Show Source): You can put this solution on YOUR website!
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
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