2. Find the value of the expression $\frac{a}{b}+\frac{b}{a}$, where $a$ and $b$ are the largest and smallest roots of the equation $x^{3}-9 x^{2}+9 x=1$, respectively.
Answer: 62.

Solution. The given equation is equivalent to the following

$$
\left(x^{3}-1\right)-9\left(x^{2}-x\right)=0 \Leftrightarrow(x-1)\left(x^{2}+x+1\right)-9 x(x-1)=0 \Leftrightarrow(x-1)\left(x^{2}-8 x+1\right)=0
$$

from which $x=1$ or $x=4 \pm \sqrt{15}$. The largest root is $a=4+\sqrt{15}$, the smallest is $-b=4-\sqrt{15}$. Then

$$
\frac{a}{b}+\frac{b}{a}=\frac{4+\sqrt{15}}{4-\sqrt{15}}+\frac{4-\sqrt{15}}{4+\sqrt{15}}=\frac{(4+\sqrt{15})^{2}+(4-\sqrt{15})^{2}}{(4+\sqrt{15})(4-\sqrt{15})}=\frac{2(16+15)}{1}=62
$$