  We study an equation $Qu=g$, where $Q$ is a continuous quadratic operator
acting from one normed space to another normed space. Obviously, if $u$ is a
solution of such equation then $-u$ is also a solution. We find conditions
implying that there are no other solutions and apply them to the study of the
Dirichlet boundary value problem for the partial differential equation $u\Delta
u =g$.
