SOLUTION: The sum of the squares of two positive consecutive integers is 130. Find the integers. Only an algebraic solution will be accepted.

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Question 252827: The sum of the squares of two positive consecutive integers is 130. Find the integers. Only an algebraic solution will be accepted.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
let n= the first integer and n+1= the second
n^2+(n+1)^2=130
n^2+n^2+2n+1=130
2n^2+2n=129
there are no integer roots
Something is wrong with problem.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 2n%5E2%2B2n%2B-129+=+0) has the following solutons:

n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A2%2A-129=1036.

Discriminant d=1036 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+1036+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%282%29%2Bsqrt%28+1036+%29%29%2F2%5C2+=+7.54673846971554
n%5B2%5D+=+%28-%282%29-sqrt%28+1036+%29%29%2F2%5C2+=+-8.54673846971554

Quadratic expression 2n%5E2%2B2n%2B-129 can be factored:
2n%5E2%2B2n%2B-129+=+2%28n-7.54673846971554%29%2A%28n--8.54673846971554%29
Again, the answer is: 7.54673846971554, -8.54673846971554. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-129+%29

perhaps you mean consecutive even or consecutive odd
first =n second n+2
n^2+(n+2)^2=130
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 2n%5E2%2B4n%2B-126+=+0) has the following solutons:

n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A2%2A-126=1024.

Discriminant d=1024 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+1024+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%284%29%2Bsqrt%28+1024+%29%29%2F2%5C2+=+7
n%5B2%5D+=+%28-%284%29-sqrt%28+1024+%29%29%2F2%5C2+=+-9

Quadratic expression 2n%5E2%2B4n%2B-126 can be factored:
2n%5E2%2B4n%2B-126+=+2%28n-7%29%2A%28n--9%29
Again, the answer is: 7, -9. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B4%2Ax%2B-126+%29

that does work
we discard the negative answer -9
7^2+9^2
49 +81=130