If `alpha,betaa n dgamma` are roots of `2x 63+x^3-7=0` , then find the value of `sum(alpha/beta+beta/alpha)` .

Answer

-3

Explanation

Equation `2x^(3) + x^(2) - 7 = 0` has roots ` alpha , beta, gamma` .
`therefore alpha + beta + gamma = - 1//2 , alpha beta + alpha gamma + beta gamma = 0 , alpha beta gamma = 7//2`
`sum((alpha)/(beta)+(beta)/(alpha))(alpha )/(beta) + (beta)/(alpha) + (beta)/(gamma) + (gamma)/(beta) + (gamma)/(alpha) + (alpha )/(gamma)`
= `(1)/(beta) (alpha + gamma) + (1)/(alpha) (beta + gamma) + (1)/(gamma) (alpha + beta)`
= `(1)/(beta) (- (1)/(2) - beta) + (1)/(alpha) ( - (1)/(2) - alpha ) + (1)/(gamma) (-(1)/(2) - gamma)`
`= - (1)/(2) ((1)/(alpha ) + (1)/(beta) + (1)/(gamma)) - 1 - 1 - 1`
`(1)/(2) ((sumalpha beta)/(alpha beta gamma)) - 3`
= - 3