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We describe a Jordan-algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems. The exactness of relaxations is proved based on rank estimates. Our approach provides a unifying viewpoint on a large number of classical results related to cones of Hermitian matrices over real and complex numbers. We describe (apparently new) results related to cones of Hermitian matrices with quaternion entries and to the exceptional 27-dimensional Euclidean Jordan algebra.