Question

已知x＝根号3＋根号2，y=根号3-根号2，求x^2-xy+y^2

已知x＝根号3＋根号2，y=根号3-根号2，求(1)x^2-xy+y^2 （2）x^3y+xy^3

要求有过程

Answer 1

解：(1)直接计算：
x^2-xy+y^2=(&3+&2)^2-(&3+&2)(&3-&2)+(&3－&2)^2
＝3+2&6+2-1+3-2&6+2
=9
或者先化简，再计算：
x^2-xy+y^2＝x^2+2xy+y^2-3xy
=（x+y）^2-3xy
=(2&3)^2-3
=9
（2）.x^3y+xy^3=xy(x^2+y^2)=xy(x^2+y^2+2xy-2xy)
=xy[(x+y)^2-2xy]
=xy[12-2]
=10
此2题的关键是要记得公式(x-y)(x+y)=x^2-y^2，同时要善于变换公式，先添后减或者相反。

Answer 2

(1)x^2=5+2根号6
y^2=5-2根号6
xy=1
x^2-xy+y^2=9
(2)x^3y+xy^3=xy(x^2+y^2)
=1*(5+2根号6+5-2根号6)
=10

Answer 3

解:(1)x^2-xy+y^2
=(x+y)^2-3xy
∵x＝√3＋√2，y=√3-√2
∴x+y=2√3,xy=(√3)^2-(√2)^2=1
∴(x+y)^2-3xy
=(2√3)^2-(3×1)=9
(2)x^3y+xy^3
=xy(x^2+y^2)
=xy[(x+y)^2-2xy]
=1×[(2√3)^2-(2×1)]
=10