Question

if x = 5+2√6 find the value of √x +1/√x​

Answer 1

√x + 1/√x = +2√3 or -2√3.

Step-by-step explanation:

Given :

x = 5 + 2√6.

To find :

The value of (√x + 1/√x).

Solution :

$x \: = 5 + 2\sqrt{6} \\ \therefore \: \frac{1}{x} = \frac{1}{5 + 2 \sqrt{6} } \\ \therefore \: \frac{1}{x} = \frac{1 \times (5 - 2 \sqrt{6} )}{(5 + 2 \sqrt{6} )(5 - 2 \sqrt{6} )} \\ \therefore \: \frac{1}{x} = \frac{5 - 2 \sqrt{6} }{ {5}^{2} - {(2 \sqrt{6} )}^{2} } \\ \therefore \: \frac{1}{x} = \frac{5 - 2 \sqrt{6} }{25 - 24} = \: \: 5 - 2 \sqrt{6}$

Now,

$x + \frac{1}{x} = 5 + 2 \sqrt{6} + 5 - 2 \sqrt{6} \\ = x + \frac{1}{x} = 10$

${( \sqrt{x} + \frac{1}{ \sqrt{x} } )}^{2} = x + \frac{1}{x} + 2 \\ = {( \sqrt{x} + \frac{1}{ \sqrt{x} } )}^{2} = \: 10 + 2 = 12 \\ \therefore \: {( \sqrt{x} + \frac{1}{ \sqrt{x} } )} = \: (+ - ) \sqrt{12} \\ {( \sqrt{x} + \frac{1}{ \sqrt{x} } )} = \: ( + - )2 \sqrt{3}$

Hence, the answer.

Answer 2

Answer:

The value of √x + (1/√x) is 2√3.

Step-by-step explanation:

Given:

x = 5 + 2√6

To find:

Value of √x + (1/√x) = ?

Solution:

We have,

x = 5 + 2√6

It can be written as,

=> x = 3 + 2 + 2√6

=> x = (√3)^2 + (√2)^2 + 2 × √3 × √2

=> x = (√3 + √2)^2

=> √x = (√3 + √2)

since, 1/√x

= 1/(√3 + √2)

=> [1/(√3 + √2)]×[(√3 - √2)/(√3 - √2)]

=> [1(√3 - √2)]/[(√3 + √2)(√3 - √3)]

since,(a + b)(a - b) = a^2 - b^2

Where, a = √3 and b = √2.

=> (√3 - √2)/[(√3)^2 - (√2)^2]

=> (√3 - √2)/(3 - 2)

=> (√3 - √2)/1

=> √3 - √2

Therefore,

=> √x + (1/√x)

=> √3 + √2 + √3 - √2

=> 2√3

Hence, the value of √x + (1/√x) is 2√3.