Question

If
$x = 2 + \sqrt{3} \\ then \: find \\ \sqrt{x} + \frac{1}{ \sqrt{x} }$
​

Answer 1

Question :

 If x = 2 + √3, then find the vale of √x + 1/√x

Answer :  

Given :  

x = 2 + √3  

Let's find the value of 1/x first  

1/x = 1/( 2 + √3 )  

Rationalising the denominator

⇒ 1/x = ( 2 - √3 ) / { ( 2 + √3 )( 2 - √3 ) }  

⇒ 1/x = ( 2 - √3 ) / { 2² - ( √3 )² }  

⇒ 1/x = ( 2 - √3 ) / ( 4 - 3 )  

⇒ 1/x = ( 2 - √3 ) / 1  

⇒ 1/x = 2 - √3  

Now, find the value of x + 1/x  

⇒ x + 1/x = 2 + √3 + 2 - √3  

⇒ x + 1/x = 4  

Adding 2 on both sides  

⇒ x + 1/x + 2 = 4 + 2  

It can be written as  

⇒ ( √x )² + ( 1/√x )² + 2( √x )( 1/√x ) = 6

Since ( a + b )² = a² + b² + 2ab  

⇒ ( √x + 1/√x )² = 6  

Taking square root on both sides  

⇒ √x + 1/√x = ± √6  

∴ the value of √x + 1/√x is ± √6.