If x^2 + x - 1 = 0, find the value of x^4 - 3x^2 + 3.
The root of x^2+x-1=0 is (-1+sqrt5)/2, so plug it in and you get 2.
x^2 + x - 1 = 0
Rearrange as :
x^2 = 1 - x (1)
And
x^4 - 3x^2 + 3 =
(x^2)^2 - 3x^2 + 3 (2)
Sub (1) into (2) and we have that
(1 - x)^2 - 3 (1 - x) +3
x*2 - 2x + 1 - 3 + 3x + 3
x^2 + x - 2 + 3
( x^2 + x - 1 ) - 1 + 3
( x^2 + x - 1 ) + 2
0 + 2 =
2
