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Question
If α,β be the roots of x2−px+q=0 and α′,β′ be those of x2−p′x+q′=0 , find the value of (α−α′)2+(β−α′)2+(α−β′)2+(β−β′)2..
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Solution
α,β are the roots of x2−px+q=0⟹α+β=p,αβ=qα′,β′ are the roots of x2−p′x+q′=0⟹α′+β′=p′,α′β′=q′
(α−α′)2+(β−α′)2+(α−β′)2+(β−β′)2=2(α2+β2)+2(α′2+β′2)−2α(α′+β′)−2β(α′+β′) =2(p2−2q)+2(p′2−2q′)−2pp′=2p2+2p′2−2pp′−4q−4q′
α′,β′ are the roots of x2−p′x+q′=0⟹α′+β′=p′,α′β′=q′
(α−α′)2+(β−α′)2+(α−β′)2+(β−β′)2=2(α2+β2)+2(α′2+β′2)−2α(α′+β′)−2β(α′+β′)
=2(p2−2q)+2(p′2−2q′)−2pp′=2p2+2p′2−2pp′−4q−4q′
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