SOLUTION: find two consecutive positve intergers such that the sum of there squares is 41

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Question 92505: find two consecutive positve intergers such that the sum of there squares is
41
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Translate the word problem:
x%5E2%2B%28x%2B1%29%5E2=41 "two consecutive positive integers such that the sum of their squares is 41"

x%5E2%2Bx%5E2%2B2x%2B1=41 Foil

x%5E2%2Bx%5E2%2B2x%2B1-41=0 Subtract 41 from both sides

2x%5E2%2B2x-40=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 2%2Ax%5E2%2B2%2Ax-40=0 ( notice a=2, b=2, and c=-40)

x+=+%28-2+%2B-+sqrt%28+%282%29%5E2-4%2A2%2A-40+%29%29%2F%282%2A2%29 Plug in a=2, b=2, and c=-40



x+=+%28-2+%2B-+sqrt%28+4-4%2A2%2A-40+%29%29%2F%282%2A2%29 Square 2 to get 4



x+=+%28-2+%2B-+sqrt%28+4%2B320+%29%29%2F%282%2A2%29 Multiply -4%2A-40%2A2 to get 320



x+=+%28-2+%2B-+sqrt%28+324+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-2+%2B-+18%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-+18%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

x+=+%28-2+%2B+18%29%2F4 or x+=+%28-2+-+18%29%2F4

Lets look at the first part:

x=%28-2+%2B+18%29%2F4

x=16%2F4 Add the terms in the numerator
x=4 Divide

So one answer is
x=4



Now lets look at the second part:

x=%28-2+-+18%29%2F4

x=-20%2F4 Subtract the terms in the numerator
x=-5 Divide

So another answer is
x=-5

So our possible solutions are:
x=4 or x=-5

Since we only care about the positive numbers, our only solution is x=4


So if x=4 then...

4%2B1=5 the 2nd number is 5

So the pair of numbers is 4 and 5




Check:

4%5E2%2B5%5E2=41 plug in the pair of numbers


16%2B25=41 Square each term

41=41 Combine like terms. Since the equation is true, our answer is verified.