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Question
If α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q, prove that `alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
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Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q
`alpha+beta=-"coefficient of x"/("coefficient of "x^2)`
`=(-(-p))/1`
= p
`alphabeta="constant term"/"coefficient of "x^2`
`=q/1`
= q
we have,
`alpha^2/beta^2+beta^2/alpha^2=(alpha^2xxalpha^2)/(beta^2xxalpha^2)+(beta^2xxbeta^2)/(alpha^2xxbeta^2)`
`alpha^2/beta^2+beta^2/alpha^2=alpha^4/(beta^2alpha^2)+beta^4/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=(alpha^4+beta^4)/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=((alpha^2+beta^2)-2alpha^2beta^2)/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=([(alpha+beta)^2-2alphabeta]^2-2(alphabeta)^2)/(alphabeta)^2`
`alpha^2/beta^2+beta^2/alpha^2=([(p)^2-2q]^2-2(q)^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^2-2q]^2-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^2xxp^2-2xxp^2xx2q+2qxx2q]-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^4-4p^2q+4q^2]-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=(p^4-4p^2q+4q^2-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=(p^4-4p^2q+2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2q)/q^2+(2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
Hence, it is proved that `alpha^2/beta^2+beta^2/alpha^2" is equal to "p^4/q^2-(4p^2)/q+2`
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