4. (20 points). For which non-negative values of the parameter $a$ does the equation $\left|\frac{x^{3}-10 x^{2}+31 x-30}{x^{2}-8 x+15}\right|=(\sqrt{2 x-a})^{2}+2-2 x \quad$ have one solution?
# Solution.

Transform $\left|\frac{x^{3}-10 x^{2}+31 x-30}{x^{2}-8 x+15}\right|=(\sqrt{2 x-a})^{2}+2-2 x$

$\left|\frac{x\left(x^{2}-10 x+25\right)+6(x-5)}{(x-5)(x-3)}\right|=2 x-a+2-2 x$

$\left|\frac{(x-5)(x-3)(x-2)}{(x-5)(x-3)}\right|=2-a$

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

Answer: when $a=1, a=2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_34be784d6fd67db418cfg-2.jpg?height=717&width=685&top_left_y=1949&top_left_x=1302)

## Criteria:

| Points | Conditions for awarding |
| :--- | :--- |
| 20 points | Correct and justified solution |
| 15 points | Functions are correctly transformed, the equation is set up, a computational error not related to the domain of the function is made in its analytical solution. |
| 10 points | Functions are correctly transformed, the equation is set up, points not included in the domain of the function are not excluded in the graphical solution. |
| 5 points | Expressions defining functions are simplified, the equation is set up, but it is not solved or is solved incorrectly. |
| 0 points | Other solutions not meeting the above criteria |