Question

若x+y=2,且(x+2)(y+2)=5,求x^3+y^3的值

Answer 1

若x+y=2,且(x+2)(y+2)=5,
xy+2(x+y)+4=5;
xy=-3;
x^3+y^3
=(x+y)(x²+y²-xy)
=2((x+y)²-3xy)
=2×(4+9)
=2×13
=26;
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Answer 2

若x+y=2,且(x+2)(y+2)=5
则x=-y
(-y+2)(y+2)=5
-y^2+4=5
y^2=1
所以，y=1 ,y=-1
y=1时，x=1
那么x^3+y^3=1^3+1^3=2
y=-1时，x=3
那么，x^3+y^3=3^3+(-1)^3=27-1=26

Answer 3

x+y=2,且(x+2)(y+2)=5,求x^3+y^3的值
2(x+y) + xy + 4 = 5
xy = 5-4-4 = -3
x(2-x) =-3
x²-2x -3 = 0
(x-3)(x+1)=0
x=3 或者 -1
y= -1 或者 3 代入

x^3+y^3
= -1 + 27
= 26

Answer 4

x^3+y^3=(x+y)^3-3x^2y-3xy^2=(x+y)^3-3xy(x+y)
x+y=2
(x+2)(y+2)=xy+2(x+y)+4=5 => xy=-3
所以x^3+y^3=2^3-3*(-3)*2=8+18=26

Answer 5

（x+2）(y+2)=5
xy+2(x+y)+4=5
xy+2*2+4=5
xy=-3
(x+y)^2=x^2+y^2+2xy
x^2+y^2-6=4
x^2+y^2=10
x^3+y^3
=(x+y)(x^2-xy+y^2)=2*(`0+3)=2*13=26