If alpha and beta are the zeros of the polynomial f(x)=x²+px+q, then form polynomial who's zeros are (alpha+beata)²and(alpha-beta)².
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alpha + beta = -b/a = -p
alpha × beta = c/a = q
(alpha + beta) ^2 = (-p)^2 = p^2
(alpha - beta) ^2 = alpha^2 + beta^2 - (2 × alpha × beta)
= (alpha + beta) ^2 - 4×alpha ×beta
= p^2 - 4q
polynomial equation : x^2 - (sum of zeros)x + prodoct of zeros
sum of zeros = p^2 +p^2 - 4q = 2p^2 - 4q
product of zeros = p^2 (p^2-4q)
polynomial equation : x^2 - (2p^2-4q)x + p^2 (p^2-4q) is the required equation
alpha × beta = c/a = q
(alpha + beta) ^2 = (-p)^2 = p^2
(alpha - beta) ^2 = alpha^2 + beta^2 - (2 × alpha × beta)
= (alpha + beta) ^2 - 4×alpha ×beta
= p^2 - 4q
polynomial equation : x^2 - (sum of zeros)x + prodoct of zeros
sum of zeros = p^2 +p^2 - 4q = 2p^2 - 4q
product of zeros = p^2 (p^2-4q)
polynomial equation : x^2 - (2p^2-4q)x + p^2 (p^2-4q) is the required equation
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