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Question Number 25184 by lucky singh last updated on 05/Dec/17
if a and b are the root of the quadratic equation x^2 +2x+3 then find the value of α^2 /β+β^2 /α
ifaandbaretherootofthequadraticequationx2+2x+3thenfindthevalueofα2/β+β2/α
Commented by prakash jain last updated on 05/Dec/17
x^2 +2x+3=0  α+β=−2  αβ=3  (α^2 /β)+(β^2 /α)  =((α^3 +β^3 )/(αβ))=(((α+β)(α^2 +β^2 −αβ))/(αβ))  =(((α+β)((α+β)^2 −3αβ))/(αβ))  =((−2((−2)^2 −3×3))/3)  =((−2(−5))/3)=((10)/3)
x2+2x+3=0α+β=2αβ=3α2β+β2α=α3+β3αβ=(α+β)(α2+β2αβ)αβ=(α+β)((α+β)23αβ)αβ=2((2)23×3)3=2(5)3=103
Answered by ibraheem160 last updated on 06/Dec/17
α+β=−2  αβ=3  (α^2 /β)+(β^2 /α)=((α^3 +β^3 )/(αβ))  α^3 +β^3 =(α+β)^3 −3αβ(α+β)  ⇒(−2)^3 −3(3)(−2)  =10  ∴ ((α^3 +β^3 )/(αβ))=((10)/3)
α+β=2αβ=3α2β+β2α=α3+β3αβα3+β3=(α+β)33αβ(α+β)(2)33(3)(2)=10α3+β3αβ=103

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