1. Solve the equation $\sqrt{x^{2}-4 x+13}+1=2 x$.
Solution. The equation is equivalent to
$$
\sqrt{x^{2}-4 x+13}=2 x-1
$$
and taking squares on both sides we get
$$
x^{2}-4 x+13=4 x^{2}-4 x+1
$$
which is equivalent to $x^{2}=4$ or $x= \pm 2$.
When we take squares we may get roots that were not roots of the original equation since $x^{2}=y^{2}$ is equivalent to $x= \pm y$ and not $x=y$. Thus we have to check our plausible roots substituting them in the original equation. Indeed -2 does not satisfy the original equation while 2 does.

Answer: the only solution is 2 .