9. For what values of the parameter $\boldsymbol{a}$ does the equation $|f(x)-5|=p(x)$, where $f(x)=\left|\frac{x^{2}-10 x+25}{x-5}-\frac{x^{2}-3 x}{3-x}\right|$, $p(x)=a \quad$ have three solutions? If there is more than one value of the parameter, indicate their product in the answer.
# Solution:

Simplify $f(x)=\left|\frac{x^{2}-10 x+25}{x-5}-\frac{x^{2}-3 x}{3-x}\right|$, we get $f(x)=$ $|2 x-5|$, where $x \neq 5, x \neq 3$.

Solve the equation || $2 x-5|-5|=a$, where $x \neq 5, x \neq 3$ graphically in the system $x O a$.

The equation has three solutions when $a=4, a=5$.

The product is 20.

Answer: 20.

![](https://cdn.mathpix.com/cropped/2024_05_06_a3feddd7c418bd26b633g-4.jpg?height=617&width=622&top_left_y=428&top_left_x=1274)

## Solution for Variant #2