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Question

If α,β are the roots of the equation x2px+q=0 then the quadratic equation whose roots are
(αβ)2(α+β)(α2+β2+αβ) and (α3β2+α2β3) is where S=p[p45p2q+5q2], P=p2q2(p45p2q+4q2)

A
None of these
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B
x2Sx+P=0
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C
x2+SxP=0
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D
x2+Sx+P=0
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Solution

The correct option is B x2Sx+P=0

Given: x2px+q=0 with roots α,β.

α+β=p,αβ=q

Let A=(αβ)2(α+β)(α2+β2+αβ)

=(p24q)p(p2q)=p[p45p2q+4q2]

Let B=α2β2(α+β)=q2p

Sum of the roots =A+B=p[p45p2q+5q2]

Product of the roots =p2q2(p45p2q+4q2)

The required equation is x2Sx+P=0


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