Let F be a quadratic field. We obtain the necessary and sufficient conditions for an element of order two in the tame kernel of F to be a fourth power in the tame kernel of F. This enables us to compute the 8-rank of the tame kernel of F. In the case when F is an imaginary quadratic field with the 8-rank of K2OF=0, the explicit structure of the Tate kernel of F can be obtained by our method.