Question

The equation 3x² - 4x + 6 = 0 has the roots α and β . Find the new equation with the roots (α/β²) and (β/α²) .

Answer 1

`{ \tt{sum = \alpha + \beta = (4)/(3) }} \\ { \tt{product = \alpha \beta = (6)/(3) = 2}} \\ \\ new \: roots : \\ { \bf{sum = \frac{ \alpha }{ { \beta }^(2) } + \frac{ \beta }{ { \alpha }^(2) } }} \\ \\ { \bf{ = \frac{ { \beta }^(2) \alpha + { \alpha }^(2) \beta }{ { (\alpha \beta )}^(2) } }} \\ \\ = { \bf{ \frac{ \alpha \beta ( \alpha + \beta )}{ {( \alpha \beta )}^(2) } }} \\ \\ = { \tt{ \frac{2( (4)/(3)) }{2 {}^(2) } }} \\ \\ sum = (2)/(3) \\ \\ { \bf{product = (1)/( \alpha \beta )}} \\ = (1)/(2) \\ \\ { \boxed{ \tt{equation : {2x}^(2) + (4)/(3) x + 1 }}}`