If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then form an equation whose roots are:
$α + \frac{1}{β}, β + \frac{1}{α}$

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Solution

Since $α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then,

$α + β = - \frac{b}{a}$

And,

$α β = \frac{c}{a}$

If the roots of any equation is $α + \frac{1}{β}$ and $β + \frac{1}{α}$ , then,

$(x - (α + \frac{1}{β})) (x - (β + \frac{1}{α})) = 0$

$(x - α - \frac{1}{β}) (x - β - \frac{1}{α}) = 0$

$(\frac{β x - α β - 1}{β}) (\frac{α x - α β - 1}{α}) = 0$

$α β x^{2} - α β^{2} x - β x - α^{2} β x + α^{2} β^{2} + α β - α x + α β + 1 = 0$

$α β x^{2} - x (α β^{2} + β + α^{2} β + α) + α^{2} β^{2} + 2 α β + 1 = 0$

$α β x^{2} - x (β (α β + 1) + α (α β + 1)) + α^{2} β^{2} + 2 α β + 1 = 0$

$α β x^{2} - x ((β + α) (α β + 1)) + α^{2} β^{2} + 2 α β + 1 = 0$

$\frac{c}{a} x^{2} - x ((- \frac{b}{a}) (\frac{c}{a} + 1)) + {(\frac{c}{a})}^{2} + 2 (\frac{c}{a}) + 1 = 0$

$\frac{c}{a} x^{2} - x (- \frac{b c}{a^{2}} - \frac{b}{a}) + {(\frac{c}{a})}^{2} + 2 (\frac{c}{a}) + 1 = 0$

$\frac{c}{a} x^{2} - x (- \frac{b c - a b}{a^{2}}) + \frac{c^{2}}{a^{2}} + 2 (\frac{c}{a}) + 1 = 0$

$a c x^{2} + (b c + a b) x + c^{2} + 2 a c + a^{2} = 0$

Therefore, the required equation is $a c x^{2} + (b c + a b) x + c^{2} + 2 a c + a^{2} = 0$ .

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