SOLUTION: The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers. Find the five integers.

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Question 195930: The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers. Find the five integers.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Consecutive integers follow the form: x, x+1, x+2, x+3, etc...


So...

"The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers." translates to x%5E2%2B%28x%2B1%29%5E2%2B%28x%2B2%29%5E2=%28x%2B3%29%5E2%2B%28x%2B4%29%5E2


x%5E2%2B%28x%2B1%29%5E2%2B%28x%2B2%29%5E2=%28x%2B3%29%5E2%2B%28x%2B4%29%5E2 Start with the given equation.


x%5E2%2Bx%5E2%2B2x%2B1%2Bx%5E2%2B4x%2B4=x%5E2%2B6x%2B9%2Bx%5E2%2B8x%2B16 FOIL


3x%5E2%2B6x%2B5=2x%5E2%2B14x%2B25 Combine like terms.


3x%5E2%2B6x%2B5-2x%5E2-14x-25=0 Get all terms to the left side.


x%5E2-8x-20=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-8, and c=-20


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-8%29+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-8, and c=-20


x+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Negate -8 to get 8.


x+=+%288+%2B-+sqrt%28+64-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Square -8 to get 64.


x+=+%288+%2B-+sqrt%28+64--80+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-20%29 to get -80


x+=+%288+%2B-+sqrt%28+64%2B80+%29%29%2F%282%281%29%29 Rewrite sqrt%2864--80%29 as sqrt%2864%2B80%29


x+=+%288+%2B-+sqrt%28+144+%29%29%2F%282%281%29%29 Add 64 to 80 to get 144


x+=+%288+%2B-+sqrt%28+144+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%288+%2B-+12%29%2F%282%29 Take the square root of 144 to get 12.


x+=+%288+%2B+12%29%2F%282%29 or x+=+%288+-+12%29%2F%282%29 Break up the expression.


x+=+%2820%29%2F%282%29 or x+=++%28-4%29%2F%282%29 Combine like terms.


x+=+10 or x+=+-2 Simplify.


So the answers are x+=+10 or x+=+-2

Since the problem mentions that the numbers are positive. So this means that the only solution is x+=+10


This means that the numbers are: 10, 11, 12, 13, and 14