Question

If x=(7+root5)/2 then find the value of x^2+1/x^2·

Answer 1

x = ( 7 + √5)/2

then,

1/x = 2/(7 +√5)

=2( 7 -√5)/44

=(7 -√5)/22

x ^2 + 1/x ^2 =?

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

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

={ (7+√5)/2+(7-√5)/22}^2 -2

={ (77 +11√5 +7 -√5)/22}^2 -2

={ 84 + 10√5}^2/484 -2

={(84 + 10√5)^2 -968} /484

Answer 2

(X+1/X)^2=X^2+1/X^2+2
((7+√5)/2+2/(7+√5))^2-2=X^2+1/X^2

((7+√5)^2+4/2(7+√5))^2-2
(49+5+14√5+4/2(7+√5))^2-2
(58+14√5/14+2√5)^2-2
(4344+1624√5)/(216+28√5)-2
(4344+1624√5-432-56√5)/ 216+28√5
(3912+1568√5)/(216+28√5)