Question

If x= 1-√5, then x⁴-4x³+4x²=?​

Answer 1

Answer:

Method 1.

x4–4x3+4x2=(x−1)4–2x2+4x−1=(x−1)4−2(x−1)2+1=25−10+1=16.■  

Method 2.

The golden ratio  ϕ=1+5√2  is the positive root of the equation  t2−t−1=0 . The other root is its conjugate:  1−5√2=x2 . Therefore  (x2)2=x2+1 , or  x2=2x+4 , and so  x4–4x3+4x2=(x2–2x)2=16 .  ■