If $3+\sqrt{5}$ is a root of the equation \[x^2 - 3x + b = 0,\]compute $b.$
We do not know that $b$ is rational, so we cannot conclude that the radical conjugate of $3+\sqrt{5},$ or $3-\sqrt{5},$ must also be a root of the equation. Instead, we turn to Vieta's formulas: the sum of the roots of the equation is $3,$ so the other root of the equation must be $3 - (3+\sqrt5) = -\sqrt5.$ Then $b$ equals the product of the roots: \[b = -\sqrt5(3+\sqrt5) = \boxed{-3\sqrt5-5}.\]Alternatively, since $3 + \sqrt{5}$ is a root of the equation,
\[(3 + \sqrt{5})^2 - 3(3 + \sqrt{5}) + b = 0.\]Hence, $b = \boxed{-3 \sqrt{5} - 5}.$