Question

if x = 3+2√2 find √x +1/√x​

Answer 1

Answer:

2√2

Step-by-step explanation:

Given x = 3 + 2√2

⇒ x = 2 + 1 + 2√2 × 1

⇒ x = (√2)2 + 12 + 2 × √2 × 1

⇒ x = (√2 + 1)2

∴ √x = (√2 + 1)

⇒ 1/√x = 1/(√2 + 1)

Multiply and divide with (√2 – 1)  

1/√x = (√2 – 1)/[(√2 – 1)(√2 + 1)]  

        = (√2 – 1)/(2 – 1)

∴  1/√x = (√2 – 1)

Now consider, (√x + 1/√x) = (√2 + 1) + (√2 – 1)

Therefore, (√x + 1/√x) = 2√2

Hope so it will help :))

Answer 2

Answer :-

x + 1/√x

Explanation : -

Given :

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

To find :

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

Solution :

$x = 3 + 2 \sqrt{3} \\ \\ above \: equation \: can \: be \: written \: as \\ \\ x = {( \sqrt{2}) }^{2} + 2( \sqrt{2} ) + {1}^{2}$

We know that,

$\boxed{\sf{ {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}}}$

$here \: a = \sqrt{2} \: and \: b = 1 \\ \\ so \: x = {( \sqrt{2} + 1) }^{2}$

$now \: substitute \: x \: \: in \: \sqrt{x} + \frac{1}{ \sqrt{x} }$

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

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

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

$= \frac{{( \sqrt{2} + 1)}^{2} + 1}{ \sqrt{2} + 1}$

$\boxed{\tt{ {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}}}$

$here \: a = \sqrt{2} \: and \: b = 1 \\ \\ so \: substitute \: the \: values \: in \: the \\ above \: formula$

$= \frac{ {( \sqrt{2})}^{2} + 2( \sqrt{2})(1) + {1}^{2} + 1}{ \sqrt{2} + 1}$

$= \frac{ {( \sqrt{2})}^{2} + 2( \sqrt{2})(1) + 1 + 1}{ \sqrt{2} + 1}$

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

$= \frac{4 + 2 \sqrt{2} }{ \sqrt{2} + 1 }$

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

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

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

$= 2 \sqrt{2}$

$\boxed{\sf{\bold{\sqrt{x} + \frac{1}{ \sqrt{x} }=2 \sqrt{2} }}}$

Identities used :-

(a + b)² = a² + 2ab + b²

Extra info :-

What is an identity ?

An equation is called an identity if it is satisfied by any value that replaces its variables.

Some Important identities :-

1] (a + b)² = a² + 2ab + b²

2](a - b)² = a² - 2ab + b²

3] (a + b)(a - b) = a² - b²

4] (x + a)(x + b) = x² + (a + b)x + ab