Question

已知x=根号5-2分之根号5-2,y=根号5+2分之根号5-2. 求（1）x+y （2）xy （3）x+y分之x²+3xy+y²

已知x=根号5-2分之根号5+2,y=根号5+2分之根号5-2. 求（1）x+y （2）xy （3）x+y分之x²+3xy+y²。 （2和-2都在根号外）

Answer 1

x=(√5+2)/(√5-2)
=(√5+2)²/[(√5-2)(√5+2)]
=(9+4√5)/(5-2)
=(9+4√5)/3

y=(√5-2)/(√5+2)
=(√5-2)²/[(√5-2)(√5+2)]
=(9-4√5)/(5-2)
=(9-4√5)/3
(1)
x+y=(9+4√5)/3+(9-4√5)/3
=(9+9)/3
=6

(2)
xy=(√5+2)/(√5-2)*(√5-2)/(√5+2)=1
(3)
(x²+3xy+y²)/(x+y)
=[(x+y)²+xy]/(x+y)
=(6²+1)/6
=37/6
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追问

x， （√5-2）（√5+2)怎么是（5-2）

追答

x=(√5+2)/(√5-2)
=(√5+2)²/[(√5-2)(√5+2)]
=(9+4√5)/(5-4)
=9+4√5

y=(√5-2)/(√5+2)
=(√5-2)²/[(√5-2)(√5+2)]
=(9-4√5)/(5-4)
=9-4√5
(1)
x+y=(9+4√5)+(9-4√5)
=9+9
=18
(3)
(x²+3xy+y²)/(x+y)
=[(x+y)²+xy]/(x+y)
=(18²+1)/18
=325/18

Answer 2

已知x=根号5-2分之根号5+2=√5(√5+2)/[(√5-2)*(√5+2)]+2=5+2√5+2=7+2√5
y=根号5+2分之根号5-2=√5(√5-2)/[(√5-2)(√5+2)]-2=5-2√5-2=3-2√5
(1) x+y=7+2√5+3-2√5=10
(2) xy=(7+2√5)(3-2√5)=1+8√5
(3) (x^2+3xy+y^2)/(x+y)=[(x+y)^2+xy]/(x+y)=[100+1+8√5]/10=(101+8√5)/10