<s>[INST] An expression involving the roots of a quadratic polynomial I have the equation $$ 9x^2 - 11x + 1 = 0 $$ whose two roots are $ \alpha $ and $ \beta $ . 
I need to evaluate $$ \frac 1 {(9\alpha-11)^2} + \frac{11\beta - 1} 9$$
What I've tried


*

*Expanded the denominator and add them, but nothing simplifies and I get even a complex expression.

*I found that $ \alpha + \beta = \frac{11}{9} 
\implies \alpha = \frac{11}{9} - \beta $


How to evaluate the value of the expression in an easier way?
 [/INST] Since $9\alpha^2-11\alpha+1=0=9\beta^2-11\beta+1$, by Vieta's theorem we have 
$$ \frac{1}{(9\alpha-11)^2}+\frac{(11\beta-1)}{9} = \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=\left(\frac{11}{9}\right)^2-2\cdot\frac{1}{9}=\color{red}{\frac{103}{81}}$$
</s>