1. Calculate $\lg 2$ and $\lg 5$ if it is known that $\lg 2 \cdot \lg 5=0.210$.
Solution. Let $\lg 2=x_{1}$ and $\lg 5=x_{2}$. Then for $x_{1}$ and $x_{2}$, we have

$$
\begin{aligned}
& x_{1} \cdot x_{2}=0.210 \\
& x_{1}+x_{2}=1
\end{aligned}
$$

so $x_{1}$ and $x_{2}$ are solutions to the quadratic equation

$$
x^{2}-x+0.210=0
$$

The solutions to this equation are 0.7 and 0.3. Since $\lg 2<\lg 5$, we get that $\lg 2=0.3$ and $\lg 5=0.7$.