Question 935085
{{{x = 7+4*sqrt(3)}}}
 
Formula:  {{{ (a+b)^2=a^2+b^2+2ab}}}
          here a = x , b =1/x
      so {{{ (x+1/x)^2= x^2+(1/x)^2+2*x*1/x}}}
             {{{ (x+1/x)^2= x^2+1/x^2+2}}}
 move 2 to the right
     {{{ (x+1/x)^2 -2 = x^2+1/x^2}}}....eq(1)
but {{{ x= 7+4*sqrt(3)}}}
          {{{1/x  = 1/(7+4*sqrt(3))}}}
          we have to rationalize the above term

             multiply and divide with {{{(7-4*sqrt(3))}}}
    {{{ 1/x = (1/(7+4*sqrt(3)))*((7-4*sqrt(3))/(7-4*sqrt(3)))}}}
          {{{ 1/x = (7-4*sqrt(3))/((7+4*sqrt(3))*(7-4*sqrt(3)))}}}
   formula: {{{(p+q)*(p-q)= p^2-q^2}}}
            {{{ 1/x = (7-4*sqrt(3))/(7^2-(4*sqrt(3))^2)}}}
              {{{1/x = (7-4*sqrt(3))/(49-16*3)}}}
              {{{1/x =(7-4*sqrt(3))/(49-48)}}}
                   {{{1/x =(7-4*sqrt(3))}}}  
 {{{ x+1/x  =  7+4*sqrt(3)+7-4*sqrt(3)}}}
       {{{ x+1/x = 14}}}
     put  x+1/x = 14 in eq (1)
 {{{ 14^2-2 =x^2+1/x^2}}}
{{{196-2= x^2+1/x^2}}}
 {{{194 =x^2+1/x^2}}}
so  {{{ x^2+1/x^2 =194}}}