[tex]\displaystyle x+\frac{1}{x}=7 \\ \\ \frac{x^2+1}{x}=7 \\ \\ 7x=x^2+1 \\ \\ x^2-7x+1=0 \\ \\ \Delta=7^2-4\cdot 1\cdot 1 \\ \\ \Delta=45 \implies x_1=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-7+\sqrt{45}}{2}=\frac{7+3\sqrt5}{2} \\ \\ x_2=\frac{-b-\sqrt{\Delta}}{2a}=\frac{7-3\sqrt5}{2} \\ \\ x_1^2=\frac{49+45+42\sqrt5}{4}=\frac{94+42\sqrt5}{4} \\ \\ \frac{94+42\sqrt5+1}{94+42\sqrt5}=\boxed{\frac{95+42\sqrt5}{94+42\sqrt5}} \\ \\ x_2^2=\frac{49+45-42\sqrt5}{4}=\frac{94-42\sqrt5}{4} \\ \\ \frac{94-42\sqrt5+1}{94-42\sqrt5}=\boxed{\frac{95-42\sqrt5}{94-42\sqrt5}}[/tex]
