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This paper is devoted to the investigation on the stability for two characteristic functions $f_1\left(z\right)=z^2+p{e}^{-z\tau }+q$ and $f_2\left(z\right)=z^2+pz{e}^{-z\tau }+q$, where $p$ and $q$ are real numbers and $\tau >0$. The obtained theorems describe the explicit stability dependence on the changing delay $\tau$. Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.