1. Solve the equation $x^{2}+3+\log _{2}\left(x^{2}-4 x+6\right)=4 x$.
Solution. We write the equation in the form

$$
\log \left((x-2)^{2}+2\right)=1-(x-2)^{2}
$$

Since

$$
\log \left((x-2)^{2}+2\right) \geq \log _{2} 2=1
$$

it follows that

$$
\log \left((x-2)^{2}+2\right) \geq 1 \geq 1-(x-2)^{2}
$$

Equality is satisfied for $x-2=0$, i.e., $x=2$, which is the only solution to the equation.