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If-x-x-1-3-4-x-1-3-10-Find-x-1-3-1-x-1-3-3-

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Question Number 213791 by hardmath last updated on 16/Nov/24
If x − (x)^(1/3) − (4/( (x)^(1/3) )) = 10 Find (x)^(1/3) − (1/( (x)^(1/3) )) + 3 = ?
$$\mathrm{If}\:\:\:\mathrm{x}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{x}}\:−\:\frac{\mathrm{4}}{\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:=\:\:\mathrm{10} \\ $$$$\mathrm{Find}\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}}\:−\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:+\:\:\mathrm{3}\:\:=\:\:? \\ $$
Commented by muallimRiyoziyot last updated on 19/Nov/24
x−8=(x)^(1/3) +2+(4/( (x)^(1/3) )) ((x)^(1/3) −2)((x^2 )^(1/3) +2(x)^(1/3) +4)=(((x^2 )^(1/3) +2(x)^(1/3) +4)/( (x)^(1/3) )) (x)^(1/3) −2=(1/( (x)^(1/3) )) → (x)^(1/3) −(1/( (x)^(1/3) ))=2
$${x}−\mathrm{8}=\sqrt[{\mathrm{3}}]{{x}}+\mathrm{2}+\frac{\mathrm{4}}{\:\sqrt[{\mathrm{3}}]{{x}}} \\ $$$$\left(\sqrt[{\mathrm{3}}]{{x}}−\mathrm{2}\right)\left(\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }+\mathrm{2}\sqrt[{\mathrm{3}}]{{x}}+\mathrm{4}\right)=\frac{\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }+\mathrm{2}\sqrt[{\mathrm{3}}]{{x}}+\mathrm{4}}{\:\sqrt[{\mathrm{3}}]{{x}}} \\ $$$$\sqrt[{\mathrm{3}}]{{x}}−\mathrm{2}=\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{x}}}\:\rightarrow\:\:\:\:\:\sqrt[{\mathrm{3}}]{{x}}−\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{x}}}=\mathrm{2} \\ $$
Commented by muallimRiyoziyot last updated on 19/Nov/24
2+3=5
$$\mathrm{2}+\mathrm{3}=\mathrm{5} \\ $$
Answered by Ghisom last updated on 17/Nov/24
x^(4/3) −x^(2/3) −10x^(1/3) −4=0 (x^(2/3) −2x^(1/3) −1)(x^(2/3) +2x^(1/3) +4)=0 x^(1/3) −(1/x^(1/3) )+3=y x^(2/3) −(y−3)x^(1/3) −1=0 y−3=2 ⇒ y=5
$${x}^{\mathrm{4}/\mathrm{3}} −{x}^{\mathrm{2}/\mathrm{3}} −\mathrm{10}{x}^{\mathrm{1}/\mathrm{3}} −\mathrm{4}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}/\mathrm{3}} −\mathrm{2}{x}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}\right)\left({x}^{\mathrm{2}/\mathrm{3}} +\mathrm{2}{x}^{\mathrm{1}/\mathrm{3}} +\mathrm{4}\right)=\mathrm{0} \\ $$$$ \\ $$$${x}^{\mathrm{1}/\mathrm{3}} −\frac{\mathrm{1}}{{x}^{\mathrm{1}/\mathrm{3}} }+\mathrm{3}={y} \\ $$$${x}^{\mathrm{2}/\mathrm{3}} −\left({y}−\mathrm{3}\right){x}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}=\mathrm{0} \\ $$$${y}−\mathrm{3}=\mathrm{2}\:\Rightarrow\:{y}=\mathrm{5} \\ $$
Answered by a.lgnaoui last updated on 17/Nov/24
posons z=^3 (√x) z^3 −z−(4/z)=10 z^4 −z^2 −10z−4=0(1) z^2 +3z−1=? (1) [ (z^2 −(1/2))^2 −((17)/4)]^2 =100(z^2 −(1/2))+50 (k^2 −((17)/4))^2 =100k^2 +50(X=z^2 −(1/2)) X^4 −((17)/2)X^2 −100X^2 +((17^2 )/4^2 )−50=0 k^4 −(((217)/2))k−((511)/4^2 )=0 217^2 +511/64=((217^2 ×4+511)/(64)) 108,647 X=((217)/4)+((108.647)/2)=((217.294+217)/4) =((434.294)/4)=108.573 z^2 =109.073 z=10,443 3(√(x )) =10,443 ^3 (√x) −1/^3 (√x) +3=5,19
$$\mathrm{posons}\:\:\boldsymbol{\mathrm{z}}=^{\mathrm{3}} \sqrt{\boldsymbol{\mathrm{x}}} \\ $$$$\boldsymbol{\mathrm{z}}^{\mathrm{3}} −\boldsymbol{\mathrm{z}}−\frac{\mathrm{4}}{\boldsymbol{\mathrm{z}}}=\mathrm{10} \\ $$$$\:\:\boldsymbol{\mathrm{z}}^{\mathrm{4}} −\boldsymbol{\mathrm{z}}^{\mathrm{2}} −\mathrm{10}\boldsymbol{\mathrm{z}}−\mathrm{4}=\mathrm{0}\left(\mathrm{1}\right) \\ $$$$\boldsymbol{\mathrm{z}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{z}}−\mathrm{1}=? \\ $$$$\left(\mathrm{1}\right)\:\:\left[\:\left(\boldsymbol{\mathrm{z}}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{17}}{\mathrm{4}}\right]^{\mathrm{2}} =\mathrm{100}\left(\boldsymbol{\mathrm{z}}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{50} \\ $$$$\:\:\:\:\left(\boldsymbol{\mathrm{k}}^{\mathrm{2}} −\frac{\mathrm{17}}{\mathrm{4}}\right)^{\mathrm{2}} =\mathrm{100}\boldsymbol{\mathrm{k}}^{\mathrm{2}} +\mathrm{50}\left(\boldsymbol{\mathrm{X}}=\boldsymbol{\mathrm{z}}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\boldsymbol{\mathrm{X}}^{\mathrm{4}} −\frac{\mathrm{17}}{\mathrm{2}}\boldsymbol{\mathrm{X}}^{\mathrm{2}} −\mathrm{100}\boldsymbol{\mathrm{X}}^{\mathrm{2}} +\frac{\mathrm{17}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} }−\mathrm{50}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{k}}^{\mathrm{4}} −\left(\frac{\mathrm{217}}{\mathrm{2}}\right)\boldsymbol{\mathrm{k}}−\frac{\mathrm{511}}{\mathrm{4}^{\mathrm{2}} }=\mathrm{0} \\ $$$$\:\:\mathrm{217}^{\mathrm{2}} +\mathrm{511}/\mathrm{64}=\frac{\mathrm{217}^{\mathrm{2}} ×\mathrm{4}+\mathrm{511}}{\mathrm{64}} \\ $$$$\mathrm{108},\mathrm{647} \\ $$$$\boldsymbol{\mathrm{X}}=\frac{\mathrm{217}}{\mathrm{4}}+\frac{\mathrm{108}.\mathrm{647}}{\mathrm{2}}=\frac{\mathrm{217}.\mathrm{294}+\mathrm{217}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{434}.\mathrm{294}}{\mathrm{4}}=\mathrm{108}.\mathrm{573} \\ $$$$\mathrm{z}^{\mathrm{2}} =\mathrm{109}.\mathrm{073}\:\:\:\:\:\:\mathrm{z}=\mathrm{10},\mathrm{443} \\ $$$$\mathrm{3}\sqrt{\mathrm{x}\:}\:\:\:=\mathrm{10},\mathrm{443}\:\:\:\: \\ $$$$ \\ $$$$\:\:\:^{\mathrm{3}} \sqrt{\mathrm{x}}\:−\mathrm{1}/^{\mathrm{3}} \sqrt{\mathrm{x}}\:+\mathrm{3}=\mathrm{5},\mathrm{19} \\ $$$$\:\: \\ $$$$\:\: \\ $$
Commented by Ghisom last updated on 17/Nov/24
this is wrong
$$\mathrm{this}\:\mathrm{is}\:\mathrm{wrong} \\ $$
Commented by a.lgnaoui last updated on 17/Nov/24
(√x) =((10,443)/3) (√x) −(1/( (√x)))+3=5,193 (there is no unique solution)
$$\sqrt{\mathrm{x}}\:=\frac{\mathrm{10},\mathrm{443}}{\mathrm{3}} \\ $$$$\sqrt{\mathrm{x}}\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}+\mathrm{3}=\mathrm{5},\mathrm{193}\:\:\left(\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{unique}\right. \\ $$$$\left.\:\mathrm{solution}\right) \\ $$

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