Question

if x= 1/3+2√2 , find the value of x-1/x​

Answer 1

Answer: if x= 1/3+2√2 , find the value of x-1/x​

Let √x - 1/√x = a

Squaring both the sides,

x + 1/x - 2 = a^2  

Putting the value,

3–2√2 + 1/(3–2√2) - 2 = a^2

a^2 = 1 - 2√2 + 1/(3–2√2)

= [(3–2√2) (1–2√2) + 1] / 3–2√2

= {3 - 8√2 + 9} / 3–2√2

= [12 - 8√2] / 3–2√2

Rationalising both the sides

= {(12 - 8√2)(3+2√2)} ÷ (9–8)

= 36 + 24√2 - 24√2 + 16(2)

= 36 - 32

=> 4

a^2 = 4

             [ OR]

((√x)-(1/√x))^2=x+(1/x)-2*x*(1/x).I.e

(a-b)^2=a*a+b*b-2*a*b.

((√x)-(1/√x))^2=(3–2√2)+(1/3–2√2)-2*(3–2√2)*(1/3–2√2).

Rationalising the above factor (3–2√2) I.e

Multiplying with (3+2√2) in numerator and denominator.so the result be (3+2√2).

((√x)-(1/√x))^2=(3–2√2)+(3+2√2)-2.

((√x)-(1/√x))^2=3+3–2=4.

Applying square root on both sides

(√x)-(1/√x)=2.

So the answer is 2.