Question

If x =`[sqrt5 - 2 ]/[ sqrt5 + 2]` and y = `[ sqrt5 + 2]/[ sqrt5 - 2 ]`; find :
x²

Answer 1

x² = `[( sqrt5 - 2 )/( sqrt5 + 2 )]^2 = [ 5 + 4 - 4sqrt5 ]/[ 5 + 4 + 4sqrt5] = [ 9 - 4sqrt5 ]/[ 9 + 4sqrt5 ]`

= `[ 9 - 4sqrt5 ]/[ 9 + 4sqrt5 ] xx [( 9 - 4sqrt5 )/( 9 - 4sqrt5 )] = (9 - 4sqrt5)^2/[(9)^2 - (4sqrt5)^2]` 

= `[ 81 + 80 - 72sqrt5]/[ 81 - 80 ] = 161 - 72sqrt5 `