4. (20 points) For what values of the parameter \( a \) does the equation \(\frac{\sqrt{x-1} \cdot\left(\left|x^{2}-10 x+16\right|-a\right)}{a x^{2}-7 x^{2}-10 a x+70 x+21 a-147}=0\) have three solutions?
# Solution.

Transform $\frac{\sqrt{x-1} \cdot\left(\left|x^{2}-10 x+16\right|-a\right)}{a x^{2}-7 x^{2}-10 a x+70 x+21 a-147}=$

$=\frac{\sqrt{x-1} \cdot\left(\left|x^{2}-10 x+16\right|-a\right)}{x^{2}(a-7)-10 x(a-7)+21(a-7)}=$

$=\frac{\sqrt{x-1} \cdot\left(\left|x^{2}-10 x+16\right|-a\right)}{(a-7)(x-7)(x-3)}$

Solve the equation graphically

$\frac{\sqrt{x-1} \cdot\left(\left|x^{2}-10 x+16\right|-a\right)}{(a-7)(x-7)(x-3)}=0$ in the system $x O a$.

$$
\left\{\begin{array}{c}
x=1 \\
\llbracket a=\left|x^{2}-10 x+16\right| \\
x \geq 1 \\
x \neq 7, x \neq 3, a \neq 7
\end{array}\right.
$$

Answer: when $a=0, a=5, a=9$ the equation has three roots.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ae08a41583c767d264cg-08.jpg?height=1105&width=717&top_left_y=1258&top_left_x=1115)

## 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 |