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Question

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

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Solution

From the given equation, we know
α+β=ba and αβ=ca

So, (α+β)2=b2a2
α2+2αβ+β2=b2a2
α2+β2=b2a22αβ
α2+β2=b2a22ca
α2+β2=b22aca2 ...(I)

Now, αβ+βα=(α)2+(β)2α.β

=b22aca2ca ....from (I)
=b22acac

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