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If `alpha` and `beta` are the zeros of the quadratic polynomial `f(x)=x^2-px+q`, prove that `(alpha^2)/beta^2+(beta^2)/(alpha^2)=p^4/q^2-4p^2/q+2`

Text Solution

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` (alpha)^2/(beta)^2 +(beta)^2/(alpha)^2= ((alpha)^4+(beta)^4)/(alphabeta)^2`
`((alpha)^4+(beta)^4)=((alpha)^2+(beta)^2)^2-2(alphabeta)^2`
`(alpha)^2+(beta)^2=(alpha+beta)^2-2alphabeta=p^2-2q`
`((alpha)^2+(beta)^2)^2-2(alphabeta)^2=(p^2-2q)^2-2q^2`
`(p^2-2q)^2-2q^2=p^4+2q^2-4p^2q;(alphabeta)^2=q^2`
`((alpha)^4+(beta)^4)/(alphabeta)^2=(p/q)^4+2-4p^2/q`
hence proved.
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