Question

If x2-3x+1=0 show that x5+1/x5=123. (x raised to 5)

Answer 1

Method 1 : Using Quadratic formula

x2−3x+1=0

x=−(−3)±(−3)2−4∗1∗1√2∗1

x=3±5√2


For x =3+5√2

x5+1x5 =(3+5√2)5+1(3+5√2)5

x5+1x5 =(3+5√)532+32(3+5√)5

Let us find (3+5–√)5=(3+5–√)2(3+5–√)3

=(9+5+65–√)(27+55–√+95–√(3+5–√))

= (14+65–√)(72+325–√)

= 1008+4485–√+4325–√+960

= 1968+8805–√


So, x5+1x5 =1968+8805√32+321968+8805√

Rationalizing the denominator:

x5+1x5 =1968+8805√32+321968+8805√⋅1968−8805√1968−8805√

x5+1x5 =1968+8805√32+32(1968−8805√)19682−8802∗5

x5+1x5 =1968+8805√32+32(1968−8805√)1024[math]x5+1x5 =1968+8805√32+1968−8805√32

Answer 2

1/x = 2(3 - √5)/4, or 2(3 + √ 5)/4 [Rationalizing the ... + 1/x) = 3^5 - 5 * 3³ + 5 * 3 [ substituting the value of x found in (3) ]