  Given a $d\times d$ quasiconvex quadratic form, $d\geq 3,$ we prove that if
the determinant of its acoustic tensor is an irreducible extremal polynomial
that is not identically zero, then the form itself is an extremal quasiconvex
quadratic form, i.e. it loses its quasiconvexity whenever a convex quadratic
form is subtracted from it. In the special case $d=3,$ we slightly weaken the
condition, namely we prove, that if the determinant of the acoustic tensor of
the quadratic form is an extremal polynomial that is not a perfect square, then
the form itself is an extremal quadratic form. In the case $d=3$ we also prove,
that if the determinant of the acoustic tensor of the form is identically zero,
then the form is either an extremal or polyconvex. Also, if the determinant of
the acoustic tensor of the form is a perfect square, then the form is either
extremal, polycovex, or is a sum of a rank-one form and an extremal, whose
acoustic tensor determinant is identically zero. Here we use the notion of
extremality introduced by Milton in [\ref{bib:Mil3}]
