For the quadratic moment problem $\gamma _{ij}=\int
\bar{z}^{i}z^{j}\,d\mu (z)\,(0\leq i+j\leq 2,|i-j|\leq 1),$ 
we showed that the necessary and sufficient condition 
for the existence of representing measure $\mu $ is $%
E(1)\geq 0.$ In particular, we also obtained the equivalent
conditions for the existence of representing measure supported 
in the unit circle $\Bbb T$ and in the closed unit disc $\Bbb D.$