Official solution
x2 + 9y2 - 4x + 3 = 0
(x2 - 4x) + (9y2) + 3 = 0
(x2 - 4x + 4) + (9y2) + 3 - 4 = 0
(x - 2)2 + (3y)2 = 1
\[{{{{(x - 2)}^2}} \over {{{(1)}^2}}} + {{{y^2}} \over {{{\left( {{1 \over 3}} \right)}^2}}} = 1\] (equation of an ellipse).
As it is equation of an ellipse, x \& y can vary inside the ellipse.
So, \[x - 2 \in [ - 1,1]\] and \[y \in \left[ { - {1 \over 3},{1 \over 3}} \right]\]
x \[\in\] [1, 3] \[y \in \left[ { - {1 \over 3},{1 \over 3}} \right]\]