If x+1x=√5, find the values of x2+1x2 and x4+1x4
x+1x=√5
Square both sides,
(x+1x)2=(√5)2
Apply (a+b)2=a2+b2+2ab
x2+(1x)2+2×x×1x=5
x2+1x2+2=5
x2+1x2=5−2=3 .......... (1)
Again, Square both sides,
(x2+1x2)2=32
(x2)2+(1x2)2+2×x2×1x2=9
x4+1x4+2=9
x4+1x4=9−2=7 ........ (2)