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 

<pre>
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^2 + S^2 = 9(L + S) + 8}}} 
      {{{L^2 + S^2 = 9L + 9S + 8}}}
      {{{L^2 + S^2 - 9L - 9S - 8 = 0}}}---- eq (ii)
      {{{(4 + S)^2 + S^2 - 9(4 + S) - 9S - 8 = 0}}} ------ Substituting 4 + S for L in eq (ii)
      {{{16 + 8S + S^2 + S^2 - 36 - 9S - 9S - 8 = 0}}}
      {{{S^2 + S^2 + 8S - 9S - 9S + 16 - 36 - 8 = 0}}}______Collecting like-terms
      {{{2S^2 - 10S - 28 = 0}}}______Combining like-terms
      {{{2(S^2 - 5S - 14) = 2(0)}}})______Factoring out GCF, 2
      {{{S^2 - 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(7)}}}

L = 7 + 4 ------ Substituting 7 for S in eq (i) 
L = 11, so the larger of the 2 natural number is: {{{highlight_green(11)}}}
You can do the check!! 

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