Solubility of oxygen and silicon in the iron-alloying liquids is important for constraining the core composition over the Earth's history. In this study, we systematically simulated the free energies of Fe-O-Si ternary liquids from 3000 K, 55 GPa to 6000 K, and 330 GPa. We found that temperature and pressure have remarkable influences on the free energies, and the nonideality of mixing is important for the chemical potentials even at high temperatures. Equilibrating with SiO$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ phase, Fe-O-Si liquids significantly enhance the solubility of Si and O simultaneously with increasing temperature. Considering the secular cooling of the Earth's core, this leads to high precipitation rates of SiO$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ once it was saturated, which would efficiently drive ancient geodynamo. If the evolution of Earth's core started from an oxygen-poor composition, later incorporations of oxygen seem to be needed to reach a core composition compatible with geophysical observations.