if-and-are-the-positive-roots-of-the-eqn-x-2-px-q-0-find-the-sum-M-1-4-1-4-

Question Number 171922 by Mikenice last updated on 21/Jun/22

if α and β are the positive roots of the eqn x^2 +px+q=0, find the sum M=(α)^(1/4) +(β)^(1/4)

$${if}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{positive}\:{roots}\:{of}\:{the}\:{eqn} \\ $$$${x}^{\mathrm{2}} +{px}+{q}=\mathrm{0},\:{find}\:{the}\:{sum}\:{M}=\sqrt[{\mathrm{4}}]{\alpha}\:+\sqrt[{\mathrm{4}}]{\beta} \\ $$

Commented by kaivan.ahmadi last updated on 22/Jun/22

we now S=α+β=−p , P=αβ=q M^2 =(√α)+(√β)+2((αβ))^(1/4) on the othere hand if K=(√α)+(√(β ))then K^2 =α+β+2(√(αβ))=S+2(√P) ⇒K=(√(S+2(√P)))=(√(−p+2(√q))) so we have M^2 =K+2(P)^(1/4) =(√(−p+2(√q)))+2(q)^(1/4)

$${we}\:{now}\:{S}=\alpha+\beta=−{p}\:,\:{P}=\alpha\beta={q} \\ $$$${M}^{\mathrm{2}} =\sqrt{\alpha}+\sqrt{\beta}+\mathrm{2}\sqrt[{\mathrm{4}}]{\alpha\beta} \\ $$$${on}\:{the}\:{othere}\:{hand}\:{if}\:{K}=\sqrt{\alpha}+\sqrt{\beta\:}{then} \\ $$$${K}^{\mathrm{2}} =\alpha+\beta+\mathrm{2}\sqrt{\alpha\beta}={S}+\mathrm{2}\sqrt{{P}} \\ $$$$\Rightarrow{K}=\sqrt{{S}+\mathrm{2}\sqrt{{P}}}=\sqrt{−{p}+\mathrm{2}\sqrt{{q}}} \\ $$$${so}\:{we}\:{have} \\ $$$${M}^{\mathrm{2}} ={K}+\mathrm{2}\sqrt[{\mathrm{4}}]{{P}}=\sqrt{−{p}+\mathrm{2}\sqrt{{q}}}+\mathrm{2}\sqrt[{\mathrm{4}}]{{q}} \\ $$

Commented by Mikenice last updated on 23/Jun/22

$${thanks}\:{sir} \\ $$

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