Question

if a = 2+√3, find a²+1/a²​

Answer 1

a = 2+√3

Then,

1/a = 1/2+√3

(Rationalising 1/2+√3)

1/2+√3 * 2-√3 / 2-√3 = 2-√3

a+1/a = 2+√3 + 2-√3

a+1/a = 2+2 (√3 and -√3 got cancelled)

a+1/a = 4

Squaring LHS and RHS

(a+1/a)2 = (4)2

a2 + 2*a*1/a + 1/a = 16

a2 + 1/a2 = 16 - 2

a2 + 1/a2 = 14