Question

if alpha and beta are the zeroes of the quadratic polynomial p[x]=3x^2-6x+4,find the value of alpha/beta +beta/alpha +2[1/alpha+1/beta]+3*alpha*beta

Answer 1

$\alpha + \beta = \frac{6}{3} = 2 \\ \alpha \beta = \frac{4}{3} \\ \frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } + 2( \frac{1}{ \alpha } + \frac{1}{ \beta } ) + 3 \alpha \beta = \\ \frac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta } + 2( \frac{ \alpha + \beta }{ \alpha \beta } ) + 3 \alpha \beta \\ = \frac{( { \alpha + \beta ) }^{2} - 2 \alpha \beta + 2( \alpha + \beta ) + 3( { \alpha \beta )}^{2} }{ \alpha \beta } \\ = \frac{ {(2)}^{2} - 2( \frac{4}{3} ) + 2 \times 2 + 3( { \frac{4}{3}) }^{2} }{ \frac{4}{3} } \\ = \frac{4 - \frac{8}{3} + 4 + \frac{16}{3} }{ \frac{4}{3} } \\ \\ = \frac{8 - \frac{8}{3} + \frac{16}{3} }{ \frac{4}{3} } \\ = \frac{24 - 8 + 16}{4} \\ = \frac{40 - 8}{4} \\ = \frac{32}{4} \\ = 8 \: \: \: ans$