post từng câu một thôi bn nhìn mệt quá

3/ Ta có:

\(x+y+z=0\)

\(\Rightarrow x^2=\left(y+z\right)^2;y^2=\left(z+x\right)^2;z^2=\left(x+y\right)^2\)

\(a+b+c=0\)

\(\Rightarrow a+b=-c;b+c=-a;c+a=-b\)

\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\)

\(\Leftrightarrow ayz+bxz+cxy=0\)

Ta có:

\(ax^2+by^2+cz^2=a\left(y+z\right)^2+b\left(z+x\right)^2+c\left(x+y\right)^2\)

\(=x^2\left(b+c\right)+y^2\left(c+a\right)+z^2\left(a+b\right)+2\left(ayz+bzx+cxy\right)\)

\(=-ax^2-by^2-cz^2\)

\(\Leftrightarrow2\left(ax^2+by^2+cz^2\right)=0\)

\(\Leftrightarrow ax^2+by^2+cz^2=0\)

1/ Đặt \(a-b=x,b-c=y,c-z=z\)

\(\Rightarrow x+y+z=0\)

Ta có:

\(\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\)

\(=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2\left(x+y+z\right)}{xyz}\)

\(=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\right)=\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2\)

2/ \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{abc}\)

\(\Leftrightarrow ab+bc+ca=1\)

Ta có:

\(M=\left(1+a^2\right)\left(1+b^2\right)\left(1+c^2\right)\)

\(=\left(ab+bc+ca+a^2\right)\left(ab+bc+ca+b^2\right)\left(ab+bc+ca+c^2\right)\)

\(=\left(a+b\right)\left(a+c\right)\left(b+a\right)\left(b+c\right)\left(c+a\right)\left(c+b\right)\)

\(=\left[\left(a+b\right)\left(b+c\right)\left(c+a\right)\right]^2\)

Ta có :   \(\left(x+\sqrt{x^2+2017}\right)\left(-x+\sqrt{x^2+2017}\right)=2017\left(1\right)\)

    \(\left(y+\sqrt{y^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017\left(2\right)\)

        nhân theo vế của ( 1 ) ; ( 2 ) , ta có :

     \(2017\left(-x+\sqrt{x^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017^2\)

    \(\Rightarrow\left(-x+\sqrt{x^2+2017}\right)\left(-y+\sqrt{y^2+2017}\right)=2017\)

  rồi bạn nhân ra , kết hợp với việc nhân biểu thức ở phần trên xong cộng từng vế , cuối cùng ta đc :

     \(xy+\sqrt{\left(x^2+2017\right)\left(y^2+2017\right)}=2017\)

     \(\Leftrightarrow\sqrt{\left(x^2+2017\right)\left(y^2+2017\right)}=2017-xy\)

     \(\Leftrightarrow x^2y^2+2017\left(x^2+y^2\right)+2017^2=2017^2-2\cdot2017xy+x^2y^2\) 

       \(\Rightarrow x^2+y^2=-2xy\Rightarrow\left(x+y\right)^2=0\Rightarrow x=-y\)

  A = 2017 

 ( phần trên mk lười nên không nhân ra, bạn giúp mk nhân ra nha :)   )

2/ \(\frac{\sqrt{x-2011}-1}{x-2011}+\frac{\sqrt{y-2012}-1}{y-2012}+\frac{\sqrt{z-2013}-1}{z-2013}=\frac{3}{4}\)

\(\Leftrightarrow\frac{4\sqrt{x-2011}-4}{x-2011}+\frac{4\sqrt{y-2012}-4}{y-2012}+\frac{4\sqrt{z-2013}-4}{z-2013}=3\)

\(\Leftrightarrow\left(1-\frac{4\sqrt{x-2011}-4}{x-2011}\right)+\left(1-\frac{4\sqrt{y-2012}-4}{y-2012}\right)+\left(1-\frac{4\sqrt{z-2013}-4}{z-2013}\right)=0\)

\(\Leftrightarrow\left(\frac{x-2011-4\sqrt{x-2011}+4}{x-2011}\right)+\left(\frac{y-2012-4\sqrt{y-2012}+4}{y-2012}\right)+\left(\frac{z-2013-4\sqrt{z-2013}+4}{z-2013}\right)=0\)

\(\Leftrightarrow\frac{\left(\sqrt{x-2011}-2\right)^2}{x-2011}+\frac{\left(\sqrt{y-2012}-2\right)^2}{y-2012}+\frac{\left(\sqrt{z-2013}-2\right)^2}{z-2013}=0\)

Dấu = xảy ra khi \(\sqrt{x-2011}=2;\sqrt{y-2012}=2;\sqrt{z-2013}=2\)

\(\Leftrightarrow x=2015;y=2016;z=2017\)

3/ \(\sqrt{\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}}\)

\(=\sqrt{\frac{\left(a-b\right)^2\left(b-c\right)^2+\left(b-c\right)^2\left(c-a\right)^2+\left(a-b\right)^2\left(c-a\right)^2}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}}\)

\(=\sqrt{\frac{\left(a^2+b^2+c^2-ab-bc-ca\right)^2}{\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2}}\)

\(=|\frac{a^2+b^2+c^2-ab-bc-ca}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}|\) là số hữu tỉ

Bài 2:

a) \(\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|-6x=0\)

\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|=6x\)

Ta có: \(\left|x+1\right|\ge0;\left|x+2\right|\ge0;\left|x+4\right|\ge0;\left|x+5\right|\ge0\)

\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|\ge0\)

\(\Rightarrow6x\ge0\)

\(\Rightarrow x\ge0\)

\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|=x+1+x+2+x+4+x+5=6x\)

\(\Rightarrow4x+12=6x\)

\(\Rightarrow2x=12\)

\(\Rightarrow x=6\)

Vậy x = 6

b) Giải:

Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{x-2}{2}=\frac{y-3}{3}=\frac{z-3}{4}=\frac{2y-6}{6}=\frac{3z-9}{12}=\frac{x-2-2y+6+3z-9}{2-6+12}=\frac{\left(x-2y+3z\right)-\left(2-6+9\right)}{8}\)

\(=\frac{14-5}{8}=\frac{9}{8}\)

+) \(\frac{x-2}{2}=\frac{9}{8}\Rightarrow x-2=\frac{9}{4}\Rightarrow x=\frac{17}{4}\)

+) \(\frac{y-3}{3}=\frac{9}{8}\Rightarrow y-3=\frac{27}{8}\Rightarrow y=\frac{51}{8}\)

+) \(\frac{z-3}{4}=\frac{9}{8}\Rightarrow z-3=\frac{9}{2}\Rightarrow z=\frac{15}{2}\)

Vậy ...

c) \(5^x+5^{x+1}+5^{x+2}=3875\)

\(\Rightarrow5^x+5^x.5+5^x.5^2=3875\)

\(\Rightarrow5^x.\left(1+5+5^2\right)=3875\)

\(\Rightarrow5^x.31=3875\)

\(\Rightarrow5^x=125\)

\(\Rightarrow5^x=5^3\)

\(\Rightarrow x=3\)

Vậy x = 3

Bài 1 hình như sai đề bạn ạ?