On the number of solutions of a quadratic equation in a normed space

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 $u\Delta u =g$.