SOLUTION: The sum of the squares of three consecutive integers is 116. Find the integers.

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Question 90300This question is from textbook
: The sum of the squares of three consecutive integers is 116. Find the integers. This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Are you sure the sum is not 110? If the sum is 116, then you get a decimal answer, which is not an integer.

x%5E2%2B%28x%2B1%29%5E2%2B%28x%2B2%29%5E2=110 Start with the given expression

x%5E2%2Bx%5E2%2B2x%2B1%2Bx%5E2%2B4x%2B4=110 Foil

x%5E2%2Bx%5E2%2B2x%2B1%2Bx%5E2%2B4x%2B4-110=0 Subtract 110 from both sides

3x%5E2%2B6x-105=0 Combine like terms




Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

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

So lets solve 3%2Ax%5E2%2B6%2Ax-105=0 ( notice a=3, b=6, and c=-105)

x+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A3%2A-105+%29%29%2F%282%2A3%29 Plug in a=3, b=6, and c=-105



x+=+%28-6+%2B-+sqrt%28+36-4%2A3%2A-105+%29%29%2F%282%2A3%29 Square 6 to get 36



x+=+%28-6+%2B-+sqrt%28+36%2B1260+%29%29%2F%282%2A3%29 Multiply -4%2A-105%2A3 to get 1260



x+=+%28-6+%2B-+sqrt%28+1296+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-6+%2B-+36%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-6+%2B-+36%29%2F6 Multiply 2 and 3 to get 6

So now the expression breaks down into two parts

x+=+%28-6+%2B+36%29%2F6 or x+=+%28-6+-+36%29%2F6

Lets look at the first part:

x=%28-6+%2B+36%29%2F6

x=30%2F6 Add the terms in the numerator
x=5 Divide

So one answer is
x=5



Now lets look at the second part:

x=%28-6+-+36%29%2F6

x=-42%2F6 Subtract the terms in the numerator
x=-7 Divide

So another answer is
x=-7

So our solutions are:
x=5 or x=-7


Plug in x=5

1st #: x=5

2nd #: y=x%2B1=5%2B1=6

3rd #: z=x%2B2=5%2B2=7


Plug in x=-7

1st #: x=-7

2nd #: y=x%2B1=-7%2B1=-6

3rd #: z=x%2B2=-7%2B2=-5


So that means our 3 numbers are

5,6,7

or
-7,-6,-5