Question 1118066
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This problem can be solved in more simple way.


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Let x be the value (now unknown) exactly half-way between  {{{sqrt(A)}}}  and  {{{sqrt(B)}}},  i.e.  x = {{{(sqrt(A) + sqrt(B))/2}}}.


So then  {{{sqrt(A)}}} = x + {{{sqrt(8)/2}}} = x + {{{sqrt(2)}}},  {{{sqrt(B)}}} = x - {{{sqrt(8)/2}}} = {{{x - sqrt(2)}}}.


Then from  {{{sqrt(A)*sqrt(B)}}} = 4  you get


{{{(x+sqrt(2))*(x-sqrt(2))}}} = 4,   or


{{{x^2 - (sqrt(2))^2}}} = 4,   which is the same as


{{{x^2 - 2}}} = 4,   which implies  {{{x^2}}} = 4 + 2 = 6;  hence,  x = {{{sqrt(6)}}}.


Then  {{{sqrt(A)}}} = {{{sqrt(6) + sqrt(2)}}}  ====>  square both sides  ====>  A = {{{8 + 2*sqrt(12)}}} = {{{8 + 4*sqrt(3)}}}.

      {{{sqrt(B)}}} = {{{sqrt(6) - sqrt(2)}}}  ====>  square both sides  ====>  B =  {{{8 - 2*sqrt(12)}}} = {{{8-4*sqrt(3)}}}.
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