7. Let $x_{0}$ be the largest (real) root of the equation $x^{4}-16 x-12=0$. Evaluate $\left\lfloor 10 x_{0}\right\rfloor$.
7. Answer. 27
Solution. Since $x^{4}-16 x-12 \equiv x^{4}+4 x^{2}+4-4\left(x^{2}+4 x+4\right) \equiv\left(x^{2}-2 x-2\right)\left(x^{2}+2 x+6\right)$, we conclude that $x_{0}=1+\sqrt{3}$ and so $1+\sqrt{2.89}<x_{0}<1+\sqrt{3.24}$. Consequently, $\left\lfloor 10 x_{0}\right\rfloor=27$.