Let λ ≠ 0 be in R . If α and β are the roots of the equation x ^ 2...

$\alpha + \beta = 1 . . . \left(i\right)$

$\alpha\beta = 2\lambda . . . \left(ii\right)$
$\alpha + \gamma = \dfrac{10}{3} . . . \left(iii\right)$
$\alpha\gamma = 9\lambda . . . \left(iv\right)$
from \left(iv\right) \text{ and } \left(ii\right)$<br />$\dfrac{\gamma}{\beta} = \dfrac{9}{2} . . . $\left(v\right)$

from \left(iii\right) - $\left(i\right)$

$\gamma - \beta = \dfrac{10}{3} - 1 = \dfrac{7}{3} . . . \left(vi\right)$

from \left(v\right)$and \left(vi\right)$

$\beta = \dfrac{2}{3}, \gamma = 3$
from \left(i\right)\alpha = 1 - \beta = 1 - \dfrac{2}{3} = \dfrac{1}{3}$<br />from \left(ii\right)\lambda = \dfrac{\alpha\beta}{2} = \dfrac{\dfrac{1}{3} \times \dfrac{2}{3}}{2} = \dfrac{1}{9}$
So $\dfrac{\beta\gamma}{\lambda} = \dfrac{\dfrac{2}{3} \times 3}{\dfrac{1}{9}} = 18$