数学高手请进!!!
这个东西( ^ )是XX 的XX 次
1, 14(a^2+b^2+c^2)=(a+2b+3c)^2,求 a:b:c
2,解方程:
( 1) (5x-96)/(x-19)+(x-8)/(x-9)=(4x-19)/(x-6)+(2x-21)/(x-8)
( 2) ab/(a+b)=1/3
bc/(b+c)=1/4
ac/(a+c)=1/5
3, x-1/x=3,求(x^10+x^8+x^2+1)/(x^10+x^6+x^4+1)的值
4, 4x-3y-6z=0,2x+4y-14z=0(z不为0),
求(2x^2+3y^2+6z^2)/(x^2+5y^2+7z^2)
5 a,b,c为有理数,ab/(a+b)=1/3,bc/(b+c)=1/4,ac/(a+c)=1/5,
求abc/(ab+bc+ac)的值
6 abc=1, 证明:1/(ab+a+1)+1/(bc+b+1)+1/(ca+c+1)=1
7 a/x+b/y+c/z=0,x/a+y/b+z/c=1,
证明:x^2/a^2+y^2/b^2+z^2/c^2=1
参考答案:1.证明:
14*(a^2+b^2+c^2)=(a+2b+3c)^2
14a^2+14b^2+14c^2=a^2+4ab+4b^2+12bc+9c^2+6ac
14a^2+14b^2+14c^2-a^2-4ab-4b^2-12bc-9c^2-6ac=0
13a^2+10b^2+5c^2-4ab-12bc-6ac=0
(9a^2-6ac+c^2)+(4a^2-4ab+b^2)+(9b^2-12bc+4c^2)
(3a-c)^2+(2a-b)^2+(3b-2c)^2=0
3a-c=0
2a-b=0
3b-2c=0
即:c=3a
b=2a
所以:a:b:c=a:2a:3a=1:2:3
2.(1)(5x-96)/(x-19)+(x-8)/(x-9)=(4x-19)/(x-6)+(2x-21)/(x-8)
=>5+(-1)/(x-19)+1+(1)/(x-9)=4+(5)/(x-6)+2+(-5)/(x-8) -------拆分
=>(-1)/(x-19)+(1)/(x-9)=(5)/(x-6)+(-5)/(x-8) ---------------分步运算
=>(-10)/(x-19)(x-9)=(5)(-2)/(x-6)(x-8)
=>(x-19)(x-9)=(x-6)(x-8)
=>x^2-28x+19*9=x^2-14x+6*8
=>14x=123
x=123/14
(2)ab/a+b=1/3 :1/a+1/b=3
bc/b+c=1/4 :1/b+1/c=4
ca/c+a=1/5 :1/c+1/a=5
可以解出：1/a=2
1/b=1
1/c=3
a=1/2
b=1
c=1/3
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