In addition to other results it is proved that 1) if \(E\) is an algebra whose spectral radius \(r_E\) is a submultiplicative seminorm and \(B\) is a subspace of \(E\), then any \(r_E\)-contraction \(T\) on \(B\) (that is, there exists a number \(\alpha\in(0,1)\) such taht \(r_E(T(x)-T(y))\leqslant \alpha\, r_E(x-y)\) for all \((x,y)\in B\times B\)) has a unique fixed point in the \(r_E\)-completion \(\widetilde{(B,r_E)}\) of \((B,r_E)\) and any \(r_E\)-contraction \(T\) on \(E\) with \(r_E(T^n(\theta _E))=0\) for each \(n\in\mathbb{N}\) (here \(\theta _E\) is the zero element of \(E\)) has a unique fixed point in \(E\); 2) if \(E\) is a metrizable \(Q\)-algebra (that is, the set of quasi-invertible elements in \(E\) is open) and \(x\in E\) with \(r_E(x)<1\), then there exists a unique quasi-square root \(y\in \widetilde{E(x)}\) of \(x\) (\(E(x)\) is the subalgebra of \(E\) generated by \(x\) and \(\widetilde{E(x)}\) is its completion) such that \(r_E(y)<1\) and 3) if \(E\) is an algebraically spectral algebra (that is, the spectrum \(\text{sp}_E(x)\) of any element \(x\in E\) coincides with the set \(\{\varphi (x):\varphi\in \text{Hom} E\}\) where \(\text{Hom} E\) is the set of all characters of \(E\)) and \(x\in E\) is an element such that \(\pi _E(x^2-x)=\theta _B\) (here \(B=A/\ker{\mathcal{G}_E}\), \(\mathcal{G}_E\) the Gelfand transformation of \(E\) and \(\pi _E\) the homomorphism from \(E\) onto \(E/\mathcal{G}_E\)), then there is a unique \(y\in \text{cl}(E(x))\cap \ker{\mathcal{G}_E}\) (the closure is taken in topology defined by \(r_E\)) such that \((x+y)^2=x+y\).