If $α$ and $β$ are the roots of the equation.
$3 x^{2} - 4 x + 1 = 0$ , from a quadratic equation whose roots are $\frac{α^{2}}{β}$ and $\frac{β^{2}}{α}$ .

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Solution

Since $α, β$ are are the roots of the equation $3 x^{2} - 4 x + 1 - 0$ .
we have $α + β = \frac{4}{3}, α β = \frac{1}{3}$
Now, for the required equation, the sum of the roots $= (\frac{α^{2}}{β} + \frac{β^{2}}{α}) = \frac{α^{3} + β^{3}}{α β}$
$= \frac{(α + β)^{3} - 3 α β (α + β)}{α β} = \frac{(\frac{4}{3})^{3} - 3 \times \frac{1}{3} \times \frac{4}{3}}{\frac{1}{3}} = \frac{28}{9}$
Also, product of the roots $= (\frac{α^{2}}{β}) (\frac{β^{2}}{α}) = α β = \frac{1}{3}$
$∴$ The required equation is $x^{2} - \frac{28}{9} x + \frac{1}{3} = 0$ or $9 x^{2} - 29 x + 3 = 0$

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