Question

設方程2x²-3x-1=0的兩根為x₁和x₂，不解方程求x14+x24的值．

Answer 1

設方程2x²-3x-1=0的兩根為x₁和x₂，
由根與系數的關系知x₁+x₂=-

b

a

=

3

2

，x₁x₂=

c

a

=-

1

2

，
∵x₁⁴+x₂⁴=（x₁²+x₂²）²-2x₁²x₂²
=[（x₁+x₂）²-2x₁x₂]²-2x₁²x₂²
=[（

3

2

）²+1]²-2×

1

4

=

161

16

．