Consider the quadratic equation \[ (*)\quad x=y+B(x,x), \] B a symmetric bilinear form on a Banach space. If for each z the linear operator B(z): \(x\mapsto B(z,x)\) has only 0 as fixed point, then solutions to (*) are unique. Existence theorems for solutions of (*) are formulated in terms of the convergence of sequences \(B(z_ n)B(z_{n-1})...B(z_ 1)x\) for certain vectors x and vector seuences \(z_ 1,...,z_ n,... \).