Question 1209552
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If   x = (19 + 8√3)^(1/2),   find   {{{(x^4 - 6x^3 - 2x^2 + 18x + 23)/(x^2 - 8x + 15)}}}.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>S T E P   &nbsp;&nbsp;&nbsp;&nbsp;b y   &nbsp;&nbsp;&nbsp;&nbsp;S T E P</U>



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(1)  {{{sqrt(19 + 8sqrt(3))}}} = {{{4+sqrt(3)}}}.

         To check and to prove, simply square right side.  You will get then  {{{19+8*sqrt(3)}}}.



(2)  So, x = {{{4 + sqrt(3)}}}.  

         It implies  x-4 = {{{sqrt(3)}}},  (x-4)^2 = 3,  x^2 - 8x + 16 = 3,  x^2 - 8x + 13 = 0.



(3)  Let's calculate the denominator  x^2 - 8x + 15.

         It is  x^2 -8x + 15 = (x^2 - 8x + 13) + 2.

         The part in parentheses is zero, so  the denominator  x^2 - 8x + 15  is simply  2.



(4)  In principle, the numerator can be calculated directly, but it is computationally boring procedure.

     It is simpler to perform a long division.

           {{{(x^4 - 6x^3 - 2x^2 + 18x +23)/(x^2 - 8x +15)}}} = x^2 + 2x - 1 + {{{(38-20x)/(x^2 - 8x+ 15)}}} = 

         = x^2 + 2x - 1 + {{{(38-20x)/2}}} = (x^2 + 2x - 1) + (19 - 10x) = x^2 - 8x + 18 = (x^2 - 8x + 13) + 5.



(5)  The value in the parentheses is zero, as we established above.

     So, we get the 



<U>ANSWER</U>.  If  x = (19 + 8√3)^(1/2),  then  {{{(x^4 - 6x^3 - 2x^2 + 18x + 23)/(x^2 - 8x + 15)}}}  is equal to 5.
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Solved.