If $α$ and $β$ are the roots of the equation $4 x^{2} - 5 x + 2 = 0$ , find the equation whose roots are
$\frac{α^{2}}{β}$ and $\frac{β^{2}}{α} .$

2 x^{2} - 5 x + 16 = 0

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32 x^{2} + 5 x + 16 = 0

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2 x^{2} + 5 x + 16 = 0

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32 x^{2} - 5 x + 16 = 0

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Solution

The correct option is C $32 x^{2} - 5 x + 16 = 0$
The given equation is: $4 x^{2} - 5 x + 2 = 0$

Sum of the roots = $\frac{5}{4}$

Product of the roots = $\frac{2}{4} = \frac{1}{2}$
If the roots are $\frac{α^{2}}{β}, \frac{β^{2}}{α}$

Sum of roots = $\frac{α^{2}}{β} + \frac{β^{2}}{α}$

= $\frac{α^{3} + β^{3}}{α β}$

= $\frac{{(α + β)}^{3} - 3 α β (α + β)}{α β}$

= $\frac{{(\frac{5}{4})}^{3} - 3 (\frac{5}{4}) (\frac{1}{2})}{\frac{1}{2}}$

= $\frac{125 - 120}{32}$

= $\frac{5}{32}$

Product of roots = $(\frac{α^{2}}{β}) (\frac{β^{2}}{α})$

= $α β$

= $\frac{1}{2}$

Hence.the equation in the standard form, $x^{2} - S x + P = 0$ can be written as:

= $x^{2} - \frac{5}{32} x + \frac{1}{2} = 0$

= $32 x^{2} - 5 x + 16 = 0$

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