  This article studies closed G2-structures satisfying the quadratic condition,
a second-order PDE system introduced by Bryant involving a parameter $\lambda.$
For certain special values of $\lambda$ the quadratic condition is equivalent
to the Einstein condition for the metric induced by the closed G2-structure
(for $\lambda = 1/2$), the extremally Ricci-pinched (ERP) condition (for
$\lambda=1/6$), and the condition that the closed G2-structure be an eigenform
for the Laplace operator (for $\lambda = 0$). Prior to the work in this
article, solutions to the quadratic system were known only for $\lambda = 1/6,$
$-1/8,$ and $2/5,$ and for these values only a handful of solutions were known.
  In this article, we produce infinitely many new examples of ERP
G2-structures, including the first example of a complete inhomogeneous ERP
G2-structure and a new example of a compact ERP G2-structure. We also give a
classification of homogeneous ERP G2-structures. We provide the first examples
of quadratic closed G2-structures for $\lambda = -1,$ $1/3,$ and $3/4,$ as well
as infinitely many new examples for $\lambda = -1/8$ and $2/5.$ Our
constructions involve the notion of special torsion for closed G2-structures, a
new concept that is likely to have wider applicability.
  In the final section of the article, we provide two large families of
inhomogeneous complete steady gradient solitons for the Laplacian flow, the
first known such examples.
