\(f(x) = 2x^2 + 11x +5\)
\(\alpha + \beta = \frac{-11}{2}\)
\(\alpha\beta = \frac52\)
(a) \(\alpha^2 + \beta^2\)
\(=( (\alpha ^2 + \beta^2) - 2\alpha\beta)^2 - 2(\alpha\beta)^2\)
\( =\left(\left( \frac{-11}{2}\right)^2 - 2.\frac52\right)^2 - 2\left(\frac52\right)^2\)
\(= \left(\frac{121}{4} - 5\right)^2 - 2 \times \frac{25}4\)
\(= (\frac{101}{4})^2 - \frac{25}2\)
\(= \frac{10201}{16} - \frac{25}{2}\)
\(= \frac{10201-200}{2}\)
\(= \frac{10001}{2}\)
(b) \(\frac1\alpha+ \frac1\beta- 2\alpha\beta\)
\(= \frac{\alpha + \beta}{\alpha\beta}- 2\alpha\beta\)
\(= \frac{\frac{-11}2}{\frac52} - 2. \frac52\)
\(= \frac{-11}{5}-5\)
\(=\frac{-11-25}{5} \)
\(= \frac{-36}{5}\)
