Question

show that x^2+1/x^2=34 if x=3+2√2
using the identities:
(a+b)^2=a^2+b^2+2ab
(a-b)^2=a^2+b^2-2ab
(a+b)(a-b)=a^2-b^2

Answer 1

given x=3+2root2

x^2+1/x^2. it is in the form of a^2+b^2

we know that (a+b)^2=a^2+b^2+2ab

a^2+b^2=(a+b)^2-2ab

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

=(3+2root2+1/3+2root2)^2-2

=(3+2root2)^2+1/3+2root2-2

=18+12 root2/3+2root2-2

=6(3+2root2)/3+2root2-2

=6-2=4