If $α, β$ are the roots of quadratic equation $a x^{3} + b x + c = 0$ , then find the value of $\frac{α}{β} + \frac{β}{α}$ .

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Solution

From the given equation, we know
$α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$

So, $(α + β)^{2} = \frac{b^{2}}{a^{2}}$
$\Rightarrow α^{2} + 2 α β + β^{2} = \frac{b^{2}}{a^{2}}$
$\Rightarrow α^{2} + β^{2} = \frac{b^{2}}{a^{2}} - 2 α β$
$\Rightarrow α^{2} + β^{2} = \frac{b^{2}}{a^{2}} - 2 \frac{c}{a}$
$\Rightarrow α^{2} + β^{2} = \frac{b^{2} - 2 a c}{a^{2}}$ ...(I)

Now, $\frac{α}{β} + \frac{β}{α} = \frac{(α)^{2} + (β)^{2}}{α . β}$

$= \frac{\frac{b^{2} - 2 a c}{a^{2}}}{\frac{c}{a}}$ ....from (I)
$= \frac{b^{2} - 2 a c}{a c}$

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If α,β are the roots of quadratic equation ax3+bx+c=0, then find the value of αβ+βα.

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If $α, β$ are the roots of quadratic equation $a x^{3} + b x + c = 0$ , then find the value of $\frac{α}{β} + \frac{β}{α}$ .