If $α$ and $β$ are the roots of $x^{2} + 5 x + 4 = 0$ , then the equation whose roots are $\frac{α + 2}{3}$ and $\frac{β + 2}{3}$ , is

9 x^{2} + 3 x + 2 = 0

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9 x^{2} - 3 x - 2 = 0

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9 x^{2} + 3 x - 2 = 0

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9 x^{2} - 3 x + 2 = 0

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Solution

The correct option is B $9 x^{2} + 3 x - 2 = 0$
Given, $α$ and $β$ are the roots of the equation $x^{2} + 5 x + 4 = 0$

$∴ α + β = - 5$ and $α β = 4$

Now,
$\frac{α + 2}{3} + \frac{β + 2}{3} = \frac{α + β + 4}{3}$

$= \frac{- 5 + 4}{3} = - \frac{1}{3}$

and $(\frac{α + 2}{3}) (\frac{β + 2}{3}) = \frac{α β + 2 (α + β) + 4}{9}$

$= \frac{4 + 2 (- 5) + 4}{9} = - \frac{2}{9}$

$∴$ Required equation is

$x^{2} - (sum of roots) x + product of roots = 0$

$\Rightarrow x^{2} + \frac{1}{3} x - \frac{2}{9} = 0$

$\Rightarrow 9 x^{2} + 3 x - 2 = 0$

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