Question

if x=2+√3 find the value of x+1/x​

Answer 1

Given

$\sf \to \: x = 2 + \sqrt{3}$

To find

$\sf \to\: x + \dfrac{1}{x}$

If

$\sf \to \: x = 2 + \sqrt{3}$

then

$\sf \to \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} }$

Now Rationalize the Denominator

$\sf \to \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\sf \to \: \dfrac{1 }{x} = \dfrac{2 - \sqrt{3} }{(2 + \sqrt{3})(2 - \sqrt{3}) }$

$\sf \to \: \dfrac{1}{x} = \dfrac{2 - \sqrt{3} }{2 {}^{2} - (\sqrt{3}) {}^{2} }$

$\sf \to \: \dfrac{1}{x} = \dfrac{2 - \sqrt{3} }{4 - {3}} = 2 - \sqrt{3}$

Now we have to find

$\sf \to \: x + \dfrac{1}{x}$

Put the value on equation

$\tt \to \: 2 + \sqrt{3} + 2 - \sqrt{3}$

$\sf \to \: 4$

Answer is 4