Question

if 2+√3 = x, find x^2 + 1/x^2

Answer 1

x = 2 +$\sqrt{3}$

so we have to find the value of $x^{2} + \frac{1}{ x^{2} }$

it is in the form of $a^{2}+ b^{2} = (a+b) ^{2} - 2ab$

so,

      $(x+ \frac{1}{x} ) ^{2}$ - 2 $(x( \frac{1}{x}))$

    = $(\frac{ x^{2}+1 }{x} ) ^{2} -2$

    now substitute x value 

=   $( \frac{(2+ \sqrt{3} ) ^{2} +1}{2+ \sqrt{3} } ) ^{2} - 2$

= $(\frac{8+ \sqrt{3} }{2 +\sqrt{3} }) ^{2} -2$

= $( \frac{4(2+ \sqrt{3}) }{2+ \sqrt{3} } ) ^{2} -2$

=$4 ^{2} -2$

= 16-2 

= 14