If $α, β$ are the roots of $2 x^{2} + 5 x + 1 = 0$ , then the equation whose roots are $2 α + 1, 2 β + 1$ is

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

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

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

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

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Solution

The correct option is C $x^{2} + 3 x - 2 = 0$
$f (x) = 2 x^{2} + 5 x + 1 = 0 \to α, β$ are the roots.
Let $y = 2 x + 1$ , where $x = α, β$
$x = \frac{y - 1}{2}$
The equation whose roots are $2 α + 1, 2 β + 1$ is,
$f (\frac{x - 1}{2}) = 0$
$\Rightarrow \frac{2 (x - 1)^{2}}{4} + \frac{5 (x - 1)}{2} + 1 = 0$
$\Rightarrow (x - 1)^{2} + 5 (x - 1) + 2 = 0 \Rightarrow x^{2} + 3 x - 2 = 0$

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