If 3x² - 5x + 1 = 0, then the value of $$\left( {{x^2} + \frac{1}{{9{x^2}}}} \right)$$  is:

Solution (By Examveda Team)

$$\eqalign{ & 3x\left( {x - \frac{5}{3} + \frac{1}{{3x}}} \right) = 0 \cr & x + \frac{1}{{3x}} = \frac{5}{3} \cr & {x^2} + \frac{1}{{9{x^2}}} + \frac{2}{3} = \frac{{25}}{9} \cr & {x^2} + \frac{1}{{9{x^2}}} = \frac{{25 - 6}}{9} \cr & {x^2} + \frac{1}{{9{x^2}}} = \frac{{19}}{9} \cr & {x^2} + \frac{1}{{9{x^2}}} = 2\frac{1}{9} \cr} $$