SOLUTION: Find two consecutive positive integers such that the sum of their squares is 85.

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Question 102548: Find two consecutive positive integers such that the sum of
their squares is 85.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Consecutive integers follow the form: x, x%2B1, x%2B2, ...., etc.


Which means their squares are x%5E2, %28x%2B1%29%5E2, etc

So the sum of their squares is:

x%5E2%2B%28x%2B1%29%5E2=85


x%5E2%2Bx%5E2%2B2x%2B1=85 Foil


2x%5E2%2B2x-84=0 Subtract 85 from both sides

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 2%2Ax%5E2%2B2%2Ax-84=0 ( notice a=2, b=2, and c=-84)




x+=+%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



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



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



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



x+=+%28-2+%2B-+26%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



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

So now the expression breaks down into two parts

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

Lets look at the first part:

x=%28-2+%2B+26%29%2F4

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

So one answer is
x=6



Now lets look at the second part:

x=%28-2+-+26%29%2F4

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

So another answer is
x=-7

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


So that means our first number is either 6 or -7. If the first number is 6, then the second number is 7


Check:

6%5E2%2B7%5E2=36%2B49=85 Since the two squares add to 85, our answer is verified.

If the first number is -7, then the second number is -6


Check:

%28-7%29%5E2%2B%28-6%29%5E2=49%2B36=85 Since the two squares add to 85, our answer is verified.



So our two numbers could be
6 and 7

or

-7 and -6