[tex]x^2+ \frac{1}{x^2} =x^2+2\cdot x\cdot \frac{1}{x} +(\frac{1}{x})^2-2\cdot x\cdot \frac{1}{x} =(x+\frac{1}{x})^2-2\\\\x=2+ \sqrt{3} \ \ \ \Rightarrow\ \ \ x+ \frac{1}{x} =2+ \sqrt{3}+ \frac{1}{2+ \sqrt{3}} =2+ \sqrt{3}+ \frac{2- \sqrt{3} }{(2+ \sqrt{3})(2- \sqrt{3})} =\\\\=2+ \sqrt{3} + \frac{2- \sqrt{3} }{4-3} =2+ \sqrt{3} +2- \sqrt{3} =4\\\\x+ \frac{1}{x} =4\ \ \ \Rightarrow\ \ \ (x+\frac{1}{x})^2-2=4^2-2=16-2=14[/tex]
