Question

Find the value of x^(2)+(1)/(x^(2)) if x-(1)/(x)=sqrt(3) .​

Answer 1

Hey mate!

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Given :

$x = 2 - \sqrt{3}$

To find :

$(x - \frac{1}{x} ) {}^{3}$

Solution :

$\begin{gathered}x = 2 - \sqrt{3} \\ \\ \frac{1}{x} = \frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } \\ \\ \frac{1}{x} = \frac{2 + \sqrt{3} }{(2) {}^{2} - ( \sqrt{3} ) {}^{2} } \\ \\ \frac{1}{x} = \frac{2 + \sqrt{3} }{4 - 3} \\ \\ \frac{1}{x} = 2 + \sqrt{3} \end{gathered}$

Now,

$\begin{gathered}x - \frac{1}{x} = 2 - \sqrt{3} - 2 - \sqrt{3} \\ \\ x - \frac{1}{x} = - 2\sqrt{3} \end{gathered}$

And,

$\begin{gathered}(x - \frac{1}{x} ) {}^{3} \\ \\ - (2 \sqrt{3} ) {}^{3} \\ \\ \therefore \boxed {\bold{- 24 \sqrt{3}}}\end{gathered}$

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Thanks for the question !

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