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

Explanation

Given that `alpha and beta` are the zeroes of polynomial `f(x) = x^2 - px + q`.

Sum of roots = `alpha + beta = p`

Product of roots = `alpha beta = q`

`therefore (alpha^2)/(beta^2)+(beta^2)/(alpha^2) = (alpha^4 + beta^4)/(alpha^2 beta^2)`

`= [(alpha^2 + beta^2)^2 – 2 alpha^2 beta^2]/(alpha beta)^2`

`= [((alpha + beta)^2 - 2 alpha beta)^2 – 2 (alpha beta)^2]/(alpha beta)^2`

`= [(p^2 - 2q)^2 - 2q^2]/q^2`

`= (p^4 + 4q^2 - 4p^2q - 2q^2)/q^2`

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