Okay… you need to find a common denominator first…

so …. x^2 is the same as x^4 / x^2

x^2 + (1/x^2) = x^4 / x^2 + 1/x^2 = (x^4 + 1) / x^2

okay… so….

#########################

# x^2 + (1/x^2) = (x^4 + 1) / x^2 #

########################

x^2 = [(sqrt 2) – 1]^2 = [(sqrt 2) – 1] [(sqrt 2) – 1]

** use the FOIL method to expand… figure out the products of first, outer, inner, last terms… and then sum up… **

x^2 = (sqrt2)(sqrt2) + (sqrt 2)(-1) + (-1)(sqrt 2) + (-1)(-1)

** see what I’m doing? **

x^2 = 2 -sqrt 2 -sqrt 2 + 1 = (3 – 2 sqrt 2)

so….

###############

# x^2 = (3-2sqrt2) #

##############

x^4 = (x^2)(x^2) = (3 – 2sqrt 2)(3 – 2sqrt 2)

** again… use FOIL method to expand…. **

x^4 = (3)(3) + (3)(-2sqrt 2) + (-2sqrt 2)(3) + (-2sqrt 2)(-2sqrt 2)

x^4 = 9 – 6sqrt 2 – 6 sqrt 2 + (4)(2)

x^4 = 9 – 12 sqrt 2 + 8

###################

# x^4 = (17 – 12 sqrt 2) #

##################

Remember we said….

x^2 + (1/x^2) = x^4 / x^2 + 1/x^2 = (x^4 + 1) / x^2 ???

…. and we found that x^2 = (3 – 2 sqrt 2)

…. and we also found that x^4 = 17 – 12 sqrt 2

** so substitute (3 – 2 sqrt 2) for x^2 in the denominator … and substitute (17 – 12 sqrt 2) for the x^4 in the numerator… like this… **

x^2 + (1/x^2) = (x^4 + 1)/x^2 = (17 – 12 sqrt 2 + 1)/(3 – 2 sqrt 2)

** combining 17 and 1… you get 18 in the numerator … **

x^2 + (1/x^2) = (18 – 12 sqrt 2) / (3 – 2sqrt 2)

** guess what?…. you can factor out a “6” from both 18 and 12 in the numerator… like this **

x^2 + (1/x^2) = 6(3 – 2sqrt 2) / (3 – 2 sqrt 2)

** Lastly, the (3 – 2sqrt 2)’s cancel out with each other…. leaving you with “6” **

So …. x^2 + (1/x^2) = 6

EDIT: When you see “^1/2” it’s the same as squareroot

x^2 + (1/x^2) with x=2^1/2 – 1

(2^1/2 – 1)^2 + 1/(2^1/2 -1)^2

(2^1/2 – 1)^2 is equal to (2^1/2 – 1) x (2^1/2 -1) This comes out to 2 – 2^1/2 – 2^1/2 + 1 which simplifies to 3 – (2 x 2^1/2) So the first half of the equation is

3 – (2 x 2^1/2) this is also the denominator in the second half

So the entire equation is

3 – (2 x 2^1/2) + [1/(3 – (2 x 2^1/2))]

Sorry but I forget how to figure out square roots without a calculator but this should help you I think.