Summary: We use the quadratic rank-one convex envelope \(qr(f)\) for \(f:M^{N\times n} \to\mathbb{R}\) to study conditions for equality of semiconvex envelopes and use the corresponding quadratic rank-one convex hull \(qr(K)\) for compact sets \(K\subset M^{N\times n}\) to give a condition for equality of semiconvex hulls. We show that \(L_c (K)=C(K)\) if and only if \(qr(K)=C(K)\). We also establish that for a function \(f\) bounded below by certain quadratic functions, \(R(f)=C(f)\) if and only if \(qr(f)=C(f)\). In particular, when \(\min\{N,n\} =2\), \(R(f)= C(f)\) if and only if \(P(f)=C(f)\).