Question

If x = 2 + √3 find x + 1x

Answer 1

Answer:

4

Step-by-step explanation:

see image for explanation

Answer 2

Question :

If x = 2 + √3 find x + 1/x

To find :

x + 1/x

Solution :

So we have the value of x

But we need to get 1/x

$\sf \small \frac{1}{x} \: is \: nothing \: but \: \frac{1}{2 + \sqrt{3} } \: because \: x \: is \: in \: the \: form \: of \: denominator$

So after that we need to apply the concept of rationalising th denominator

Rationalising :

• To rationalize the denominator, you must multiply both the numerator and the denominator by the conjugate of the denominator. Remember to find the conjugate all you have to do is change the sign between the two terms.

• Distribute both the numerator and the denominator.

So rationalising 1/x we get

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

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

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

Note : We need to use the formula of (a + b)(a - b) = a² - b² to get the denominator

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

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

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

So after that we will solve x + 1/x

$x + \frac{1}{x}$

Putting the values we get

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

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

$4$

Final Answer :

$\sf So \: the \: value \: of \: x + \frac{1}{x} is \: \boxed{ \purple{4}}$