If a, ß are the roots of the equation x2 + px + p + q = 0, then find the value of a^2 + aB+B^2+q.​

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1. Answer:

   α,β be the roots of the equation x2+px+p2+q=0…(1).

   We know that..

   An equation formed by the two roots α,β is..

   x2−(α+β)x+αβ=0…(2)

   Now compairing (1) & (2) ,we get…

   α+β=−p

   & αβ=p2+q

   The term α2+αβ+β2+q…(3) can be written as…

   (α2+β2)+αβ+q

   =(α+β)2–2αβ+αβ+q

   =(α+β)2−αβ+q

   =(−p)2−(p2+q)+q

   =p2−p2−q+q

   =0

   Hence the required value of α2+αβ+β2+q is 0

   Problem is done.

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