Question

If x=3+2under root 2 then find the value of (under root x - 1/under root x)

Answer 1

Answer :

√x - 1/√x = 2

Solution :

• Given : x = 3 + 2√2
• To find : √x - 1/√x = ?

We have ,

x = 3 + 2√2

Thus ,

1/x = 1/(3 + 2√2)

Now ,

Rationalising the denominator of the term in RHS , we get ;

=> 1/x = (3 - 2√2) / (3 + 2√2)(3 - 2√2)

=> 1/x = (3 - 2√2) / [ 3² - (2√2)² ]

=> 1/x = (3 - 2√2) / (9 - 8)

=> 1/x = (3 - 2√2) / 1

=> 1/x = 3 - 2√2

Also ,

We know that ,

(A - B)² = A² + B² - 2AB

Thus ,

If A = √x and B = 1/√x , then

=> (√x - 1/√x)² = (√x)²+ (1/√x)²- 2•√x•(1/√x)

=> (√x - 1/√x)² = x + 1/x - 2

=> (√x - 1/√x)² = 3 + 2√2 + 3 - 2√2 - 2

=> (√x - 1/√x)² = 3 + 3 - 2

=> (√x - 1/√x)² = 4

=> √x - 1/√x = √4

=> √x - 1/√x = 2

Hence ,

√x - 1/√x = 2

[ For alternative method , please refer to the attachment . ]

Answer 2

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

And,

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \\ \\ = > \frac{3 - 2 \sqrt{2} }{ {(3)}^{2} - {(2 \sqrt{2} )}^{2} } \\ \\ = > \frac{3 - 2 \sqrt{2} }{9 - 8} \\ \\ \huge \boxed{\frac{1}{x} = 3 - 2 \sqrt{2} }$

Thus,

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