# How to calculate quantiles of sums?

Let $$X_1, X_2$$ be independent normal distributions. Consider the quantile $$x$$ such that $$P(X_1 + X_2 \le x) = \alpha$$ for some $$\alpha \in (0,1)$$.

My question is, how does this quantity relate to the quantity $$x_1, x_2$$ where $$x_1$$ satisfies $$P(X_1 \le x_1) = \alpha$$ and $$x_2$$ satisfies $$P(X_2 \le x_2) = \alpha$$?

Is $$x = x_1 + x_2$$? Is it bigger? Is it less? What is the general result for general values of $$\alpha$$ and specific choices of the distribution parameters?