Question

If x=2+√3 then find the value of x^2 – 1/x^2​

Answer 1

Hello!

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x = 2 + √3

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

Now,

$x = 2 + \sqrt{3} \\ \\ \bf squaring \: both \: sides : \\ \\ x {}^{2} = (2 + \sqrt{3} ) {}^{2} \\ \\ x {}^{2} = 4 + 3 + 4 \sqrt{3} \\ \\ x {}^{2} = 7 + 4 \sqrt{3}$

And,

$\frac{1}{x} = 2 - \sqrt{3} \\ \\ \bf squaring \: both \: sides : \\ \\ \frac{1}{x {}^{2} } = (2 - \sqrt{3} ) {}^{2} \\ \\ \frac{1}{x {}^{2} } = 4 + 3 - 4 \sqrt{3} \\ \\ \frac{1}{ x{}^{2} } = 7 - 4 \sqrt{3}$

Now,

$x {}^{2} - \frac{1}{x {}^{2} } \\ \\ 7 + 4 \sqrt{3} -( 7 - 4 \sqrt{3} ) \\ \\ 7 + 4 \sqrt{3} - 7 + 4 \sqrt{3} \\ \\ 8 \sqrt{3}$

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Answer : 8√3