If x² - 3x + 1 = 0, then the value of $${x^2} + x + \frac{1}{x} + \frac{1}{{{x^2}}}$$    is?

Solution (By Examveda Team)

$$\eqalign{ & {x^2} - 3x + 1 = 0 \cr & \Rightarrow {x^2} + 1 = 3x \cr & \Rightarrow x + \frac{1}{x} = 3 \cr & {\text{Squaring both sides}} \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + 2 = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 7 \cr & \therefore {x^2} + x + \frac{1}{x} + \frac{1}{{{x^2}}} \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + x + \frac{1}{x} \cr & \Rightarrow 7 + 3 \cr & \Rightarrow 10 \cr} $$