• # question_answer If$\alpha \And \beta$be the roots of the quadratic equation ${{x}^{2}}+2x+3=0$then the value of $\frac{{{\alpha }^{2}}}{\beta }+\frac{{{\beta }^{2}}}{\alpha }$is A) $\frac{10}{3}$                         B) $\frac{3}{10}$            C) $\frac{2}{3}$                          D) None of these

[a] ${{x}^{2}}+2x+3=0$ $\therefore \alpha +\beta =-2\And \alpha .\beta =3$ Now $\frac{{{\alpha }^{2}}}{\beta }+\frac{{{\beta }^{2}}}{\alpha }=\frac{{{\alpha }^{3}}+{{\beta }^{2}}}{\alpha .\beta }=\frac{{{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta )}{\alpha .\beta }$ $=\frac{{{(-2)}^{3}}-3.3(-2)}{3}=\frac{-8+18}{3}=\frac{10}{3}$ Hence, option [a] is correct.