If $α, β$ are the roots of the equation $2 x^{2} + 5 x + 6 = 0$ , then find the equation whose roots are $\frac{1}{α}$ and $\frac{1}{β}$ .

$5 x^{2} - 6 x + 2 = 0$

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$5 x^{2} + 6 x + 2 = 0$

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$6 x^{2} + 5 x + 2 = 0$

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$6 x^{2} - 5 x + 2 = 0$

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Solution

The correct option is C
$6 x^{2} + 5 x + 2 = 0$

Let's assume $x_{0}$ is a root of the equation $2 x^{2} + 5 x + 6 = 0$ , we want to find the equation whose root is $\frac{1}{x_{0}}$ .

To find the equation whose roots are $\frac{1}{x_{0}}$ , we replace $x$ by $\frac{1}{x}$
$\Rightarrow 2 \times {(\frac{1}{x})}^{2} + 5 \times \frac{1}{x} + 6 \Rightarrow 2 + 5 x + 6 x^{2} = 0$

Alternate method:
We can also solve this by finding the sum and product of the roots.

$α + β = - \frac{5}{2}$
$α β = \frac{6}{2}$
$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = - \frac{5}{6}$
$\frac{1}{α β} = \frac{2}{6}$

New quadratic equation with roots $\frac{1}{α}$ and $\frac{1}{β}$ is
$x^{2} - x (\frac{1}{α} + \frac{1}{β}) + (\frac{1}{α} \frac{1}{β}) = 0$
$\Rightarrow x^{2} + \frac{5}{6} x + \frac{2}{6} = 0$
$\Rightarrow 6 x^{2} + 5 x + 2 = 0$

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If α,β are the roots of the equation 2x2+5x+6=0, then find the equation whose roots are 1α and 1β.

If $α, β$ are the roots of the equation $2 x^{2} + 5 x + 6 = 0$ , then find the equation whose roots are $\frac{1}{α}$ and $\frac{1}{β}$ .