9.1. Given positive numbers $p$ and $r$. Let $f(x)$ and $g(x)$ be linear functions with roots $p$ and $r$. Find all roots of the equation $f(x) g(x)=f(0) g(0)$.
Answer. $x_{1}=0, x_{2}=p+r$.

First solution. Let the given functions be of the form: $f(x)=a x+b$ and $g(x)=c x+d$. Then the equation becomes $(a x+b)(c x+d)-b d=0$, that is, $x(a c x+a d+b c)=0$. One root of this equation is $x_{1}=0$, and the second is $x_{2}=\frac{-a d-b c}{a c}$, which means $x_{2}=-\frac{d}{c}-\frac{b}{a}$. It remains to note that $-\frac{d}{c}$ is $r$, and $-\frac{b}{a}$ is $p$.

Second solution. Write our functions in the form: $f(x)=a(x-p), g(x)=c(x-r)$. Then the equation becomes

$$
a c(x-p)(x-r)-a c p r=0
$$

that is, $a c x(x-p-r)=0$, from which the answer follows.

Comment. It is only proven that one of the roots is $0$ - 2 points.

The answer depends on variables not specified in the condition - no additional points are awarded.