Question

X=9+4√5, then find (√x-1/√x)

Answer 1

Step-by-step explanation:

x = 9+4√5

x = 4+5+4√5

= 2² + √5² + 2.2.√5

= ( 2 + √5)²

√x = 2 + √5

1/√x = 1/(2+√5)=(2-√5)/(2²-√5²) =(2-√5)/(-1)= √5 - 2

√x - 1/√x

2+√5 - (√5-2)

2+2

4

Answer 2

Answer:

Given,

→ x = 9 + 4√5

Writing 9 as 4 + 5

→ x = 4 + 5 + 4√5

4 can be written as 2² and 5 as (√5)², as well as 4√5 is 2*2*√5

→ x = 2² + (√5)² + 2*2*√5

The above expression meets with a² + b² + 2ab, so writing that as ( a + b )²

→ x = ( 2 + √5 )²

→ √x = 2 + √5 --- (1)

→ 1/√x = 1/(2+√5)

$\implies \dfrac{1}{\sqrt{x}} =\dfrac{1}{2+\sqrt5}\times\dfrac{2-\sqrt5}{2-\sqrt5}\\\\\\\implies\dfrac{1}{\sqrt{x}} =\dfrac{2-\sqrt5}{2^2 - (\sqrt5)^2}\\\\\\\implies\dfrac{1}{\sqrt{x}}=\dfrac{2-\sqrt5}{4-5}\\\\\\\implies\dfrac{1}{\sqrt{x}} =\sqrt5 - 2$

Hence, from (1)

√x - 1/√x

= 2 + √5 - ( √5 - 2 )

= 2 + √5 - √5 + 2

= 2 + 2

= 4