jmc

An $x$-intercept is a point on the graph that lies on the $x$-axis, so $y=0$. When $y=0$, $x=3$, so $a=3$.

A $y$-intercept is a point on the graph that lies on the $y$-axis, so $x=0$. Hence, the $y$-intercepts correspond to the real roots of the quadratic equation $2y^2-6y+3=0$. By Vieta's formulas, the sum of the roots of this quadratic is $6/2=3$, so $b+c=3$.

Therefore, $a+b+c=3+3=\boxed{6}$.