if-x-1-3-x-36-1-3-3-find-x-1-x-

Question Number 154385 by amin96 last updated on 17/Sep/21

i f \sqrt[3]{x} - \sqrt[3]{x - 36} = 3 f i n d x - \frac{1}{x}

Answered by amin96 last updated on 18/Sep/21

x = t^{3} \Rightarrow t - \sqrt[3]{t^{3} - 36} = 3 \Rightarrow \sqrt[3]{t^{3} - 36} = t - 3

t^{3} - 36 = t^{3} - 9 t^{2} + 27 t - 27 \Rightarrow t^{2} - 3 t - 1 = 0

t - \frac{1}{t} = 3 t^{2} + \frac{1}{t^{2}} = 11 \Rightarrow (t^{2} + \frac{1}{t^{2}}) (t - \frac{1}{t}) = 33

t^{3} - t + \frac{1}{t} - \frac{1}{t^{3}} = 33 \Rightarrow t^{3} - \frac{1}{t^{3}} = 36 x - \frac{1}{x} = 36

Answered by liberty last updated on 18/Sep/21

I f \sqrt[3]{x} - \sqrt[3]{x - 36} = 3 t h e n

f i n d x - \frac{1}{x} .

s o l u t i o n :

\Rightarrow 3 + \sqrt[3]{x - 36} - \sqrt[3]{x} = 0

\Rightarrow 27 + x - 36 - x = - 3 \times 3 \sqrt[3]{x (x - 36)}

\Rightarrow - 9 = - 9 \sqrt[3]{x (x - 36)}

\Rightarrow 1 = x^{2} - 36 x

\Rightarrow \frac{1}{x} = x - 36 \Rightarrow x - \frac{1}{x} = 36

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