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Question Number 70617 by Rio Michael last updated on 06/Oct/19

given that α and β are roots of the equation x^2 −5x + 4 =0 α>0 and β >0 find an equation whose roots are (√α) and (√β) how do i find (√(α )) + (√β)

$${given}\:{that}\:\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\:{the}\:{equation}\:{x}^{\mathrm{2}} −\mathrm{5}{x}\:+\:\mathrm{4}\:=\mathrm{0}\: \\ $$ $$\alpha>\mathrm{0}\:{and}\:\beta\:>\mathrm{0} \\ $$ $${find}\:{an}\:{equation}\:{whose}\:{roots}\:{are}\:\sqrt{\alpha}\:{and}\:\sqrt{\beta}\: \\ $$ $$ \\ $$ $${how}\:{do}\:{i}\:{find}\:\:\sqrt{\alpha\:}\:+\:\sqrt{\beta}\: \\ $$

Answered by mr W last updated on 06/Oct/19

αβ=4 ⇒(√α)(√β)=2 α+β=5 ⇒((√α))^2 +((√β))^2 =5 ⇒((√α))^2 +((√β))^2 +2(√( α))(√β)=5+2(√α)(√β) ⇒((√α)+(√β))^2 =5+2×2=9 ⇒(√α)+(√β)=(√9)=3 the eqn. with roots (√α) and (√β) is x^2 −3x+2=0

$$\alpha\beta=\mathrm{4}\:\Rightarrow\sqrt{\alpha}\sqrt{\beta}=\mathrm{2} \\ $$ $$\alpha+\beta=\mathrm{5}\: \\ $$ $$\Rightarrow\left(\sqrt{\alpha}\right)^{\mathrm{2}} +\left(\sqrt{\beta}\right)^{\mathrm{2}} =\mathrm{5} \\ $$ $$\Rightarrow\left(\sqrt{\alpha}\right)^{\mathrm{2}} +\left(\sqrt{\beta}\right)^{\mathrm{2}} +\mathrm{2}\sqrt{\:\alpha}\sqrt{\beta}=\mathrm{5}+\mathrm{2}\sqrt{\alpha}\sqrt{\beta} \\ $$ $$\Rightarrow\left(\sqrt{\alpha}+\sqrt{\beta}\right)^{\mathrm{2}} =\mathrm{5}+\mathrm{2}×\mathrm{2}=\mathrm{9} \\ $$ $$\Rightarrow\sqrt{\alpha}+\sqrt{\beta}=\sqrt{\mathrm{9}}=\mathrm{3} \\ $$ $${the}\:{eqn}.\:{with}\:{roots}\:\sqrt{\alpha}\:{and}\:\sqrt{\beta}\:{is} \\ $$ $${x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}=\mathrm{0} \\ $$

Commented byRio Michael last updated on 06/Oct/19

thank yiu so much mr W

$${thank}\:{yiu}\:{so}\:{much}\:{mr}\:{W} \\ $$

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