We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are “optimal” in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any n ≥ 2 48 n\geq 2^{48} such that n n is either odd or divisible by 2 9 2^9 , any infinite free Burnside group with exponent n n is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.