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Question

If α,β are the roots of the equation ax2+bx+c=0, then the value of (aα2+c)(aα+b)+(aβ2+c)(aβ+b) is

A
b(b22ac)4a
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B
b24ac2a
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C
b(b22ac)a2c
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D
none of these
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Solution

The correct option is D b(b22ac)a2c
aα2+c
=a(α2+ca)
=a(α2+α.β)
=aα(α+β)
=aα(ba)
=bα.
aα+b
=a(α+ba)
=a(ααβ)
=aβ
Similarly
aβ2+c
=bβ
and aβ+b
=aα
Taking the ratio, we get
ba(αβ+βα)
=ba(β2+α2α.β)
=ba(b22acac)
=b(b22ac)a2c

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