Niveau de r\'epartition des polyn\^omes quadratiques et crible majorant pour les entiers friables

We obtain new estimates on the level of distribution of the set $\{Q(n)\}$ where $Q\in{\mathbb Z}[X]$ is irreducible quadratic, for well-factorable moduli, improving a result due to Iwaniec. As a by-product of our arguments, we study the Chebyshev problem of estimating $\max\{P^+(n^2-D), n\leq x\}$ and make explicit, in Deshouillers-Iwaniec's state-of-the-art result, the dependence on the Selberg eigenvalue conjecture. Combined with the construction of an upper-bound sieve for numbers free of large factors, we obtain new upper bounds for the quantity $\Psi_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leq y\}|$ for $Q\in{\mathbb Z}[X]$ linear or quadratic.