2. Determine the real numbers $x, y, z$ for which

$$
x+y+z-21=2 \sqrt{x-4}+4 \sqrt{y-9}+6 \sqrt{z-22}
$$
Solution:

we look for perfect squares that contain the three radicals

![](https://cdn.mathpix.com/cropped/2024_06_07_954bb18b226b938f15feg-2.jpg?height=498&width=1691&top_left_y=2023&top_left_x=176)
$(y-9)-2 \cdot 2 \sqrt{y-9}+4=(\sqrt{y-9}-2)^{2}$ ..... $1 \text{p}$
$-4+1-9+4-22+9=-21$, we obtain a sum of perfect squares equal to zero $\Leftrightarrow$ each square is
$\Rightarrow \sqrt{x-4}=1, \sqrt{y-9}=2, \sqrt{z-22}=3$ ..... $1 \text{p}$
$\Rightarrow x-4=1, y-9=4, z-22=9$ ..... $1 \text{p}$
$\Rightarrow x=5, \quad y=13, z=31$ ..... $1 \text{p}$