Question

if x= (3-2√2) , show that √x-1/√x=+-2​

Answer 1

Given:-

→x= (3-2√2)

To show:-

→ (√x-1/√x)= ±2

Solution:-

Firstly,let's find the value of (√x-1/√x)².

=>(√x-1/√x)²

We know that,(a-b)²=a²+b²-2ab

= (√x)²+(1/√x)²-2×√x×1/√x

= x+1/x-2

∴(√x-1/√x)²= x+1/x-2

Here,value of x is given as 3-2√2.

So,now let's calculate the value of 1/x

=>1/x = 1/(3-2√2)

By rationalizing 1/(3-2√2),we get:-

= 1/(3-2√2)×(3+2√2)/(3+2√2)

= (3+2√2)/(3-2√2)(3+2√2)

= (3+2√2)/(3)²-(2√2)²

= (3+2√2)/9-8

= (3+2√2)/1

= 3+2√2

∴1/x = 3+2√2

Now:-

=>x+1/x-2 = (3-2√2)+(3+2√2)-2

=>x+1/x-2 = 6-2

=>x+1/x-2 = 4

=>(√x-1√x)² = 4. [∵(√x-1/√x)² = x+1/x-2]

=>(√x-1√x) = ±2

Hence,proved.