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Question Number 85198 by otchereabdullai@gmail.com last updated on 19/Mar/20

Find x if ^3 (√(x ))+(√x)=(√(12))

$$\mathrm{Find}\:\mathrm{x}\:\mathrm{if}\:\:\:\:^{\mathrm{3}} \sqrt{\mathrm{x}\:}+\sqrt{\mathrm{x}}=\sqrt{\mathrm{12}} \\ $$

Answered by MJS last updated on 19/Mar/20

let x=y^6 y^3 +y^2 −(√(12))=0 let y=z−(1/3) z^3 −(1/3)z+(2/(27))−2(√3)=0 D=4(−(1/3))^3 +27((2/(27))−2(√3))^2 >0 ⇒ ⇒ if you need an exact solution use Cardano′s formula z≈1.57613 y≈1.24280 x≈3.68470 is the only solution ∈C

$$\mathrm{let}\:{x}={y}^{\mathrm{6}} \\ $$$${y}^{\mathrm{3}} +{y}^{\mathrm{2}} −\sqrt{\mathrm{12}}=\mathrm{0} \\ $$$$\mathrm{let}\:{y}={z}−\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${z}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{3}}{z}+\frac{\mathrm{2}}{\mathrm{27}}−\mathrm{2}\sqrt{\mathrm{3}}=\mathrm{0} \\ $$$${D}=\mathrm{4}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} +\mathrm{27}\left(\frac{\mathrm{2}}{\mathrm{27}}−\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} >\mathrm{0}\:\Rightarrow \\ $$$$\Rightarrow\:\mathrm{if}\:\mathrm{you}\:\mathrm{need}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}\:\mathrm{use} \\ $$$$\mathrm{Cardano}'\mathrm{s}\:\mathrm{formula} \\ $$$${z}\approx\mathrm{1}.\mathrm{57613} \\ $$$${y}\approx\mathrm{1}.\mathrm{24280} \\ $$$${x}\approx\mathrm{3}.\mathrm{68470} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{only}\:\mathrm{solution}\:\in\mathbb{C} \\ $$

Commented by otchereabdullai@gmail.com last updated on 19/Mar/20

Thanks a lot prof MJS you are funtastic God bless you

$$\mathrm{Thanks}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{prof}\:\mathrm{MJS}\:\mathrm{you}\:\mathrm{are}\:\mathrm{funtastic}\: \\ $$$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$

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