If $α, β$ are the roots of the equation $3 x^{2} + 5 x + 4 = 0$ , then the quadratic equation whose roots are $α^{2}, β^{2}$ .

$9 x^{2} + x + 24 = 0$

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$9 x^{2} - x + 16 = 0$

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$9 x^{2} - x + 24 = 0$

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$9 x^{2} + x + 16 = 0$

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Solution

The correct option is B
$9 x^{2} - x + 16 = 0$

$α + β = \frac{- 5}{3}, α β = \frac{4}{3}$

To find the equation whose roots are $α^{2}, β^{2}$ , we have to find $α^{2} + β^{2}$ and $α^{2} β^{2}$

$α^{2} + β^{2} = (α + β)^{2} - 2 α β = \frac{25}{9} - \frac{2 \times 4}{3} = \frac{1}{9} α^{2} β^{2} = \frac{16}{9}$
The equation is :
$x^{2} - \frac{1}{9} x + \frac{16}{9} = 0$
$\Rightarrow 9 x^{2} - x + 16 = 0$

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