The role of tetraquark operators in studying the isodoublet strange $\kappa$ and isovector nonstrange $a_0(980)$ scalar mesons in lattice QCD is examined using an ensemble with $m_\pi\approx230$ MeV and spatial extent L such that $m_\pi L\approx4.4$. Hermitian correlation matrices using both single-meson, meson-meson, and tetraquark interpolating operators are used to extract the spectrum of finite-volume stationary states in the appropriate symmetry channels. Hundreds of local and extended tetraquark operators are explored. Determinations of the spectrum in each channel are found to be unreliable without the inclusion of at least one tetraquark operator. For example, the inclusion of tetraquark operators with isospin 1/2 and strangeness 1 quantum numbers reveals the existence of an additional energy level in the Kη sub-system below the $K\eta$ threshold. The implications of this on parametrizing the scattering K-matrix through a well-known quantization condition to extract properties of the $\kappa$ and $a_0(980)$ scalar meson resonances are discussed.