Question

if alpha and beta are zeros of p(x)=4x2-5x-1, then find the value of alpha2/beta+beta2/alpha.

Answer 1

$$\begin{gathered} 4x^2-5x-1=0 \\ So, \\ \alpha +\beta =\frac{5}{4} \\ \alpha \beta =\frac{-1}{4} \\ Now, \\ \frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }=\frac{{\alpha }^3+{\beta }^3}{\beta \alpha }=\frac{{\left(\alpha +\beta \right)}^3-3\alpha \beta \left(\alpha +\beta \right)}{\beta \alpha } \\ =\frac{\frac{125}{64}+{\displaystyle \frac{3}{4}}\times \frac{5}{4}}{\frac{-1}{4}}=-\frac{125+60}{16}=-\frac{185}{16} \end{gathered}$$