Question

If a = 2 + √3, then the value of a - 1/a is
2√3

Answer 1

$a = 2 + \sqrt{3}$

$\frac{1}{a} = \frac{1}{2 + \sqrt{3} }$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{(2 + \sqrt{3)} (2 - \sqrt{3)} }$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{( {2) {}^{2} - \sqrt{3 {}^{} } }^{2} }$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3}$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{1}$

$\frac{1}{a} = 2 - \sqrt{3}$

Now,

$= a - \frac{1}{a}$

$= 2 + \sqrt{3} - (2 - \sqrt{3} )$

$= 2 + \sqrt{3} - 2 + \sqrt{3}$

$= 2 \sqrt{3}$