9.2. a) Given the quadratic equation $x^{2}-9 x-10=0$. Let $a$ be its smallest root. Find $a^{4}-909 a$. b) For the quadratic equation $x^{2}-9 x+10=0$, where $b$ is the smallest root, find $b^{4}-549 b$.
Answer: a) 910; b) -710. Solution. a) Let's solve the problem in general. Let $x^{2}-c x+d=0-$ be a quadratic equation, and $a-$ be its root. Then $a^{4}=(c a-d)^{2}=c^{2} a^{2}-2 a c d+d^{2}=$ $c^{2}(c a-d)-2 a c d+d^{2}=a\left(c^{3}-2 c d\right)+d^{2}-c^{2} d$. Therefore, $a^{4}-\left(c^{3}-2 c d\right) a=d^{2}-c^{2} d$. When $c=9, d=-10$, we get $a^{4}-909 a=910$, and when $c=9, d=10$, we have $b^{4}-549 b=-710$. (In part a), the result can be easily and directly obtained).