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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-factorablemoduli, improving a result due to Iwaniec. As a by-product of our arguments, westudy the Chebyshev problem of estimating $\max\{P^+(n^2-D), n\leq x\}$ andmake explicit, in Deshouillers-Iwaniec's state-of-the-art result, thedependence on the Selberg eigenvalue conjecture. Combined with the constructionof an upper-bound sieve for numbers free of large factors, we obtain new upperbounds for the quantity $\Psi_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leqy\}|$ for $Q\in{\mathbb Z}[X]$ linear or quadratic.

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