Question

若x，y，满足x^2+y^2=5\4，xy=-1\2，求x^3+y^3

若x，y，满足x^2+y^2=5\4，xy=-1\2，求x^3+y^3

Answer 1

x²+y²=5/4，xy=-1/2，
（x+y）²=（x²+y²）+2xy=5/4-1=1/4，
x+y=1/2，或x+y=-1/2，
x³+y³=（x+y）（x²-xy+y²）=1/2×（5/4+1/2）=7/8，
或，x³+y³=（x+y）（x²-xy+y²）=-1/2×（5/4+1/2）=-7/8，

Answer 2

x²+y²=(x+y)²-2xy=5/4

(x+y)²-2×(-1/2)=5/4
(x+y)²+1=5/4
(x+y)²=1/4
x+y=±1/2
x³+y³=(x+y)(x²-xy+y²)=(x+y)[(x+y)²-3xy]

=±(1/2)(1/4+3/2)
=±(7/8)

Answer 3

正负