SOLUTION: This is from a worksheet: Find two consecutive integers such that the sum of their squares is 85. I have getten this far: n^2+(n+1)^2=85 At some point I will have to divide

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Question 88630: This is from a worksheet: Find two consecutive integers such that the sum of their squares is 85.
I have getten this far:
n^2+(n+1)^2=85
At some point I will have to divide each side by half?
Found 2 solutions by jim_thompson5910, ankor@dixie-net.com:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
n%5E2%2B%28n%2B1%29%5E2=85+

n%5E2%2Bn%5E2%2B2n%2B1=85+ Foil

n%5E2%2Bn%5E2%2B2n%2B1-85=0+ Subtract 85 from both sides

2n%5E2%2B2n-84=0+ Combine like terms


Now let's use the quadratic formula to solve for n:


Starting with the general quadratic

an%5E2%2Bbn%2Bc=0

the general solution using the quadratic equation is:

n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 2%2An%5E2%2B2%2An-84=0 ( notice a=2, b=2, and c=-84)

n+=+%28-2+%2B-+sqrt%28+%282%29%5E2-4%2A2%2A-84+%29%29%2F%282%2A2%29 Plug in a=2, b=2, and c=-84



n+=+%28-2+%2B-+sqrt%28+4-4%2A2%2A-84+%29%29%2F%282%2A2%29 Square 2 to get 4



n+=+%28-2+%2B-+sqrt%28+4%2B672+%29%29%2F%282%2A2%29 Multiply -4%2A-84%2A2 to get 672



n+=+%28-2+%2B-+sqrt%28+676+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



n+=+%28-2+%2B-+26%29%2F%282%2A2%29 Simplify the square root



n+=+%28-2+%2B-+26%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

n+=+%28-2+%2B+26%29%2F4 or n+=+%28-2+-+26%29%2F4

Lets look at the first part:

n=24%2F4 Add the terms in the numerator
n=6 Divide

So one answer is
n=6
Now lets look at the second part:

n=-28%2F4 Subtract the terms in the numerator
n=-7 Divide

So another answer is
n=-7

So our solutions are:
n=6 or n=-7


Check:
6%5E2%2B%286%2B1%29%5E2=85+ Plug in n=6

6%5E2%2B%287%29%5E2=85+ Add

36%2B49=85 Square each number

85=85 Add. So this solution works


%28-7%29%5E2%2B%28-7%2B1%29%5E2=85+ Plug in n=-7

%28-7%29%5E2%2B%28-6%29%5E2=85+ Add

49%2B36=85 Square each number

85=85 Add. So this solution works


So we have 2 pairs of numbers

6 and 7
or
-7 and -6

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Find two consecutive integers such that the sum of their squares is 85.
I have gotten this far:
n^2 + (n+1)^2 = 85
:
to continue, FOIL (n+1)(n+1)
n^2 + (n^2 + 2n + 1) = 85
:
n^2 + n^2 + 2n + 1 - 85 = 0: subtract 85 from both sides:
:
2n^2 + 2n - 84 = 0; combine like terms, you have a quadratic equation
:
n^2 + n - 42 = 0; simplify, divide equation (each term) by 2
:
(n+7)(n-6) = 0; factors easily:
:
n = -7
and
n = +6
:
Both these solutions will work, here's x = -7
(-7^2) + (-6^2) =
+49 + 36 = 85
:
:
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