Example 3. Solve the equation

$$
\sqrt{1+3 x}=x+1
$$
Solution. Squaring both sides of the equation, we get

$$
1+3 x=x^{2}+2 x+1
$$

i.e., the equation

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

which is a consequence of the original equation.

Let's find its roots: $x_{1}=0$ and $x_{2}=1$. Substituting each of the found roots into the original equation, we verify that both are roots of the original equation.

This equation serves as an example that squaring the original equation does not always lead to the appearance of extraneous roots.