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If alpha,beta are the roots of x^2+p x+q...

If `alpha,beta` are the roots of `x^2+p x+q=0a n dgamma,delta` are the roots of `x^2+r x+s=0,` evaluate `(alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta)` in lterms of `p ,q ,r ,a n dsdot` Deduce the condition that the equation has a common root.

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The correct Answer is:
`(q-s)^(2)-rqp-rsp+sp^(2)+qr^(2)`
Since, `alpha beta` are the roots of `x^(2)+px+q=0` and `lamda, delta` are the roots of `x^(2)+rx+s=0`
`therefore alpha +beta=-p, alphabeta=q`
`and gamma, delta=-r,gamma delta=s`
Now, `(alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta)`
`=[alpha^(2)-(gamma+delta)alpha+gammadelta][beta^(2)-(gamma+delta)beta+gammadelta]`
`=(alpha^(2)+ra+s)(beta^(2)+rbeta+s)`
`=(alphabeta)^(2)+r(alpha+beta)alphabeta+s(alpha^(2)+beta^(2))+alphabetar^(2)+rs(alpha+beta)+s^(2)`
`=q^(2)-rpq+s(p^(2)-2q)+qr^(2)-rsp+s^(2)`
`=(q-s)^(2)-rqp-rsp+sp^(2)+qr^(2)`
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