It seems that what you call quadratic regression is actually the minimal square error regression. In this case the computation is very easy:
1) Multiply both left sides by A'(3xn) arriving to
A'(3xn)B(nx1) = A'(3xn)A(nx3) X(3x1)
2) Now multiply both left sides by the inverse of A'(nx1) A(nx3) arriving to
inv(A'(3xn)A(nx3))A'(3xn)B(nx1) = X(3x1)
3) Now use svd to evaluate the inverse above, see Most efficient matrix inversion in MATLAB
See also Minimizing error of a formula in MATLAB (Least squares?)