If α and β are the roots of equation x² + αx + β = 0 then find α³ + β³ = ?

Solution (By Examveda Team)

$$\eqalign{ & {\text{ }}{x^2}{\text{ + }}\alpha x + \beta = 0 \cr & {\text{Sum of root}} \cr & \alpha + \beta = \frac{{ - \alpha }}{1}\,......(i) \cr & \alpha \beta = \beta \,......(ii) \cr & {\text{From (i) and (ii)}} \cr & {\text{Then, }}\alpha = 1 \cr & {\text{Then, }}\beta = - 2 \cr & {\text{Then value of }} \cr & \Leftrightarrow {\alpha ^3} + {\beta ^3} \cr & = 1 + {\left( { - 2} \right)^3} \cr & = - 7 \cr} $$