SOLUTION: The difference between 2 natural numbers is 4.The sum of their squares is greater than 9 times sum by 8.Find the numbers

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Question 825271: The difference between 2 natural numbers is 4.The sum of their squares is greater than 9 times sum by 8.Find the numbers
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
This is somewhat ambiguous, but here is a try.

x and y are the natural numbers to find.
x%2By=4;
x%5E2%2By%5E2-9%28x%2By%29=8.

The sum of just the two numbers can be substituted to give x%5E2%2By%5E2-9%2A4=8,
x%5E2%2By%5E2-36-8=0
x%5E2%2By%5E2=44
Going back to x+y=4, y=4-x, so substitute into the circle equation:
x%5E2%2B%284-x%29%5E2=44
x%5E2%2B16-8x%2Bx%5E2=44
2x%5E2-8x%2B16-44=0
2x%5E2-8x-28=0
highlight_green%28x%5E2-4x-14=0%29
'
x=%284%2B-+sqrt%2872%29%29%2F2; unless I made a mistake, this value will not be a Natural Number. Neither will be y Natural.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The difference between 2 natural numbers is 4.The sum of their squares is greater than 9 times sum by 8.Find the numbers 


Let the larger and smaller of the 2 numbers be L, and S, respectively

Then: L - S = 4______L = 4 + S ----- eq (i)

Also, L%5E2+%2B+S%5E2+=+9%28L+%2B+S%29+%2B+8 

      L%5E2+%2B+S%5E2+=+9L+%2B+9S+%2B+8

      L%5E2+%2B+S%5E2+-+9L+-+9S+-+8+=+0---- eq (ii)

      %284+%2B+S%29%5E2+%2B+S%5E2+-+9%284+%2B+S%29+-+9S+-+8+=+0 ------ Substituting 4 + S for L in eq (ii)

      16+%2B+8S+%2B+S%5E2+%2B+S%5E2+-+36+-+9S+-+9S+-+8+=+0

      S%5E2+%2B+S%5E2+%2B+8S+-+9S+-+9S+%2B+16+-+36+-+8+=+0______Collecting like-terms

      2S%5E2+-+10S+-+28+=+0______Combining like-terms

      2%28S%5E2+-+5S+-+14%29+=+2%280%29)______Factoring out GCF, 2

      S%5E2+-+5S+-+14+=+0

      (S + 2)(S - 7) = 0

      S = - 2               OR                 S = 7 

Since - 2 is NOT a natural number, the smaller of the 2 natural numbers is: highlight_green%287%29



L = 7 + 4 ------ Substituting 7 for S in eq (i) 

L = 11, so the larger of the 2 natural number is: highlight_green%2811%29

You can do the check!! 



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