Question

已知a^2-3a-1=0,求a^3+1/a^3的值

详细过程，谢谢

Answer 1

a²-1=3a
两边除以a
a-1/a=3
两边平方
a²-2+1/a²=9
a²+1/a²=11

(a+1/a)²=a²+2+1/a²=13
a+1/a=±√13

a³+1/a³
=(a+1/a)(a²-a×1/a+1/a²)
=(a+1/a)(a²-1+1/a²)
=±√13(11-1)
=±10√13

Answer 2

a^2-3a-1=0,
a-3-1/a=0
a-1/a=3

(a+1/a)^2=(a-1/a)^2+4=13
a+1/a=±√13

a^2+1/a^2=(a-1/a)^2+2=11

当a+1/a=√13时
a^3+1/a^3
=(a+1/a)(a^2-1+1/a^2)
=√13*(11-1)
=10√13

当a+1/a=-√13时
a^3+1/a^3
=(a+1/a)(a^2-1+1/a^2)
=-√13*(11-1)
=-10√13

Answer 3

解： ∵a^2-3a-1=0
∴ a- 1/a =3
∴ a + 1/a=±√(a+1/a)^2=±√[(a-1/a)^2+4]=±√13
∴a^3+1/a^3=(a+1/a)[(a- 1/a)^2+1]=±√11×（9+1）=±10√13

Answer 4

解：a^2-3a-1=0,
a-3-1/a=0
a-1/a=3

(a+1/a)^2=(a-1/a)^2+4=13
a+1/a=±√13

a^2+1/a^2=(a-1/a)^2+2=11

当a+1/a=√13时
a^3+1/a^3
=(a+1/a)(a^2-1+1/a^2)
=√13*(11-1)
=10√13

当a+1/a=-√13时
a^3+1/a^3
=(a+1/a)(a^2-1+1/a^2)
=-√13*(11-1)
=-10√13

Answer 5

y