[tex]x+\frac1{x}=8\\\frac{x^2}x+\frac1{x}=\frac{8x}x /*x\\x^2+1=8x\\x^2-8x+1=0\\\\\Delta=(-8)^2-4*1*1=64-4=60\\\sqrt{\Delta}=\sqrt{60}=2\sqrt{15}\\\\x_1=\frac{8-2\sqrt{15}}2=\frac{2(4-\sqrt{15})}{2}=4-\sqrt{15}\\x_2=\frac{8+2\sqrt{15}}2=4+\sqrt{15}[/tex]
[tex]x^2+(\frac{1}x)^2=? \\\\\text{Dla }x_1: \\\\(4-\sqrt{15})^2+(\frac{1}{4-\sqrt{15}})^2=16-8\sqrt{15}+15+\frac{1}{16-8\sqrt{15}+15}=31-8\sqrt{15}+\frac{1}{31-8\sqrt{15}}=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{(31-8\sqrt{15})(31+8\sqrt{15})}=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-960}=31-8\sqrt{15}+\frac{31+8\sqrt{15}}1=31-8\sqrt{15}+31+8\sqrt{15}=31+31=62[/tex]
[tex]\text{Dla }x_1:\\(4+\sqrt{15})^2+(\frac1{4+\sqrt{15}})^2=16+8\sqrt{15}+15+\frac{1}{16+8\sqrt{15}+15}=31+8\sqrt{15}+\frac{1}{31+8\sqrt{15}}=31+8\sqrt{15}+\frac{31-8\sqrt{15}}{(31+8\sqrt{15})(31-8\sqrt{15})}=31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-960}=31+8\sqrt{15}+31-8\sqrt{15}=31+31=62[/tex]
