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Question Number 5592 by Rasheed Soomro last updated on 21/May/16
Find the positive values of x for  which 9x^(2/3) +4x^(−(2/3)) =37 .
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{9x}^{\frac{\mathrm{2}}{\mathrm{3}}} +\mathrm{4x}^{−\frac{\mathrm{2}}{\mathrm{3}}} =\mathrm{37}\:. \\ $$
Answered by prakash jain last updated on 21/May/16
y=x^(2/3)   9y+(4/y)=37  9y^2 −37y+4=0  (9y−1)(y−4)=0  y=(1/9) or y=4  x^(2/3) =1/9  x^2 =(1/3^6 )⇒x=(1/(27))  similary second solution x=8
$${y}={x}^{\mathrm{2}/\mathrm{3}} \\ $$$$\mathrm{9}{y}+\frac{\mathrm{4}}{{y}}=\mathrm{37} \\ $$$$\mathrm{9}{y}^{\mathrm{2}} −\mathrm{37}{y}+\mathrm{4}=\mathrm{0} \\ $$$$\left(\mathrm{9}{y}−\mathrm{1}\right)\left({y}−\mathrm{4}\right)=\mathrm{0} \\ $$$${y}=\frac{\mathrm{1}}{\mathrm{9}}\:{or}\:{y}=\mathrm{4} \\ $$$${x}^{\mathrm{2}/\mathrm{3}} =\mathrm{1}/\mathrm{9} \\ $$$${x}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{6}} }\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{27}} \\ $$$$\mathrm{similary}\:\mathrm{second}\:\mathrm{solution}\:{x}=\mathrm{8} \\ $$

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