Question

1已知a+b=3,ab=-4,求a^3+b^3,a^4+b^4 2已知3(a^2+b^2+c^2)=(a+b+c)^2,求证a=b=c

要有过程哒！会的都答，不会答一题就好。。。谢谢大家

Answer 1

1.两边平方：(a+b)^2 =a^2 + 2ab + b^2 =9
∵ab=-4
∴a^2 + b^2 =9-2ab=17
a^3 + b^3 =(a+b)(a^2 - ab + b^2) =3×[17-(-4)]=63
a^4 + b^4=(a^2 + b^2)^2 - 2a^2b^2 =(17)^2 - 2×(-4)^2 =257

2.(a+b+c)^2=[(a+b)+c]^2
=(a+b)^2 + 2(a+b)c + c^2
=a^2 + 2ab + b^2 + 2ac + 2bc + c^2
=a^2 + b^2 + c^2 + 2ab + 2bc + 2ac
∵a^2 + b^2 + c^2 + 2ab + 2bc + 2ac=3a^2 + 3b^2 + 3c^2
∴2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ac=0
a^2 - 2ab + b^2 + a^2 - 2ac + c^2 + b^2 - 2bc + c^2=0
(a-b)^2 + (a-c)^2 + (b-c)^2 =0
∴(a-b)^2=0 , (a-c)^2=0 , (b-c)^2=0
∴a=b , a=c , b=c
即：a=b=c

Answer 2

a^2+b^2
=(a+b)^2-2ab
=3^2-2*(-4)
=9+8
=17

a^3+b^3,
=(a+b)(a^2-ab+b^2)
=3[(a^2+2ab+b^2)-3ab]
=3[(a+b)^2-3ab]
=3*[3^2-3*(-4)]
=3*(9+12)
=3*21
=63

a^4+b^4
=a^4+2a^2b^2+b^4-2a^2b^2
=(a^2+b^2)^2-2(ab)^2
=17^2-2*(-4)^2
=289-32
=257

Answer 3

a^3+b^3=（a+b)(a^2-ab+b^2)=(a+b)(a^2+2ab+b^2-3ab)=(a+b)【(a+b)^2-3ab】=63
a^4+b^4=(a^2+b^2)^2-2(a*b)^2=【(a+b)^2-2ab】^2-2(a*b)^2=257
要考试了。。。时间叫紧凑。。。先解一题吧。。。希望你满意~

Answer 4

已知3(a^2+b^2+c^2)=(a+b+c)^2,求证a=b=c
3a^2+3b^2+3c^2=a^2+b^2+c^2+2ab+2ac+2bc
2a^2+2b^2+2c^2=2ab+2ac+2bc
(a^2-2ab+b^2)+(a^2-2ac+c^2)+(b^2-2bc+c^2)=0
(a-b)^2+(a-c)^2+(b-c)^2=0
所以a-b=0 a-c=0 b-c=0
所以a=b=c