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.

Explanation

`:'alpha, beta` are the roots of the equation
`x^(2)+px+q=0`
`:.alpha+beta=-p,alpha beta=q`……….i
and `gamma, delta` are the roots of the equation `x^(w2)+rx+s=0`
`:.gamma+delta=-r,gamma delta=s`…ii
Now `(alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta)`
`=[(alpha^(2)-alpha(gamma+delta)+gamma delta][beta^(2)-beta(gamma+delta)+gamma delta]`
`=(alpha^(2)+r alpha +s)(beta^(2)+rbeta+s)` [from Eq (ii) ]
`impliesalpha^(2)+r alpha beta(alpha +beta)+r^(2) alpha beta+s(alpha^(2)+beta^(2))`
`+sr(alpha+beta)+s^(2)`
`=alpha^(2) beta^(2)+r alpha beta(alpha +beta)+r^(2) alpha beta+s[(alpha +beta)^(2)-2alpha beta]`
`+sr(alpha +beta)+s^(2)`
`=q^(2)-pqr+r^(2)q+s(p^(2)-2q)+sr(-p)+s^(2)`
`=(q-s)^(2)-rpq+r^(2)q+sp^(2)-prs`
`=(q-s)^(2)-rq(p-r)+sp(p-r)`
`=(q-s)^(2)+(p-r)(sp-rq)`............iii
For a common root (let `alpha=gamma` or `beta=delta`)
then `(alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta)=0` .........iv
From eqs (iii) and iv) we get
`(q-s)^(2)+(p-r)(sp-rq)=0` ltbr. `=(q-s)^(2)=(p-r)(rq-sp)` which is the required condition.