\(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}=0\Leftrightarrow\frac{bc+ac-ab}{abc}=0\)

Vì \(a,b,c\ne0\Rightarrow abc\ne0\)

\(\Rightarrow bc+ac-ab=0\)

\(\Rightarrow\hept{\begin{cases}\left(bc+ac\right)^2=\left(ab\right)^2\\\left(bc-ab\right)^2=\left(-ac\right)^2\\\left(ac-ab\right)^2=\left(-bc\right)^2\end{cases}\Rightarrow\hept{\begin{cases}b^2c^2+c^2a^2-a^2b^2=-2abc^2\\b^2c^2+a^2b^2-a^2c^2=2ab^2c\\a^2c^2+a^2b^2-b^2c^2=2a^2bc\end{cases}}}\)

\(\Rightarrow E=\frac{a^2b^2c^2}{2ab^2c}+\frac{a^2b^2c^2}{-2abc^2}+\frac{a^2b^2c^2}{2a^2bc}\)

\(\Rightarrow E=\frac{ac}{2}-\frac{ab}{2}+\frac{bc}{2}=\frac{ac-ab+bc}{2}=\frac{0}{2}=0\)

CHÚC BẠN HỌC TỐT

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\frac{bc+ac-ab}{abc}=0\)

Vì \(a,b,c\ne0\Rightarrow a.b.c\ne0\)

\(\Rightarrow bc+ac-ab=0\)

\(\Rightarrow\hept{\begin{cases}\left(bc+ac\right)^2=\left(ab\right)^2\\\left(bc-ab\right)^2=\left(-ac\right)^2\\\left(ac-ab\right)^2=\left(-bc\right)^2\end{cases}\Rightarrow}\hept{\begin{cases}b^2c^2+c^2a^2-a^2b^2=-abc^2\\b^2c^2+a^2b^2-a^2c^2=2ab^2c\\a^2c^2+a^2b^2-b^2c^2=2a^2bc\end{cases}}\)

\(\Rightarrow E=\frac{a^2b^2c^2}{2ab^2c}+\frac{a^2b^2c^2}{-2abc^2}+\frac{a^2b^2c^2}{2a^2bc}\)

\(\Rightarrow E=\frac{ac}{2}-\frac{ab}{2}+\frac{bc}{2}=\frac{ac-ab+bc}{2}=\frac{0}{2}=0\)

Vậy \(E=0\)

làm nổi à bạn. 

1. Ta có : x + y + z = 0 \(\Rightarrow\)( x + y + z )² = 0 \(\Rightarrow\)x² + y² + z² = - 2 ( xy + yz + xz )\(S=\frac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}=\frac{-2\left(xy+yz+xz\right)}{2\left(x^2+y^2+z^2\right)-2\left(yz+xz+xy\right)}\)

\(S=\frac{-2\left(xy+yz+xz\right)}{-4\left(xy+yz+xz\right)-2\left(yz+xz+xy\right)}=\frac{-2\left(xy+yz+xz\right)}{-6\left(xy+yz+xz\right)}=\frac{1}{3}\)

2. a + b + c = 1 \(\Rightarrow\)( a + b + c )² = 1 \(\Rightarrow\)a² + b² + c² + 2 ( ab + bc + ac ) = 1 \(\Rightarrow\)ab + bc + ac = 0

Áp dụng tính chất dãy tỉ số bằng nhau, ta có : \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=\frac{x+y+z}{a+b+c}=x+y+z\)

\(\Rightarrow\)x = a ( x + y + z ) ; y = b ( x + y + z ) ; z = c ( x + y + z )

Ta có : xy + yz + xz = ab ( x + y + z )² + bc ( x + y + z )² + ac ( x + y + z )² = ( x + y + z )² ( ab + bc + ac ) = 0

3. sửa đề : 3x - y = 3z

Ta có : \(\hept{\begin{cases}3x-y=3z\\2x+y=7z\end{cases}\Rightarrow\hept{\begin{cases}\left(3x-y\right)+\left(2x+y\right)=3z+7z\\2x+y=7z\end{cases}\Rightarrow}\hept{\begin{cases}5x=10z\\y=7z-2x\end{cases}\Rightarrow}\hept{\begin{cases}x=2z\\y=3z\end{cases}}}\)

\(\Rightarrow\)\(S=\frac{x^2-2xy}{x^2+y^2}=\frac{\left(2z\right)^2-2.2z.3z}{\left(2z\right)^2+\left(3z\right)^2}=\frac{4z^2-12z^2}{4z^2+9z^2}=\frac{-8z^2}{13z^2}=\frac{-8}{13}\)

mấy cái trên la a^2.b chứ không pải a tất cả mũ 2b

Có \(ab+bc+ac=abc\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\)

Áp dụng các bđt sau:Với x;y;z>0 có: \(\dfrac{1}{x+y+z}\le\dfrac{1}{9}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\) và \(\dfrac{1}{x+y}\le\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\) 

Có \(\dfrac{1}{a+3b+2c}=\dfrac{1}{\left(a+b\right)+\left(b+c\right)+\left(b+c\right)}\le\dfrac{1}{9}\left(\dfrac{1}{a+b}+\dfrac{2}{b+c}\right)\)\(\le\dfrac{1}{9}.\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{2}{b}+\dfrac{2}{c}\right)=\dfrac{1}{36}\left(\dfrac{1}{a}+\dfrac{3}{b}+\dfrac{2}{c}\right)\)

CMTT: \(\dfrac{1}{b+3c+2a}\le\dfrac{1}{36}\left(\dfrac{1}{b}+\dfrac{3}{c}+\dfrac{2}{a}\right)\)

\(\dfrac{1}{c+3a+2b}\le\dfrac{1}{36}\left(\dfrac{1}{c}+\dfrac{3}{a}+\dfrac{2}{b}\right)\)

Cộng vế với vế => \(VT\le\dfrac{1}{36}\left(\dfrac{6}{a}+\dfrac{6}{b}+\dfrac{6}{c}\right)=\dfrac{1}{36}.6\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)=\dfrac{1}{6}\)

Dấu = xảy ra khi a=b=c=3

Có \(a+b=2\Leftrightarrow2\ge2\sqrt{ab}\Leftrightarrow ab\le1\)

\(E=\left(3a^2+2b\right)\left(3b^2+2a\right)+5a^2b+5ab^2+2ab\)

\(=9a^2b^2+6\left(a^3+b^3\right)+4ab+5ab\left(a+b\right)+20ab\)

\(=9a^2b^2+6\left(a+b\right)^3-18ab\left(a+b\right)+4ab+5ab\left(a+b\right)+20ab\)

\(=9a^2b^2+48-18ab.2+4ab+5.2.ab+20ab\)

\(=9a^2b^2-2ab+48\)

Đặt \(f\left(ab\right)=9a^2b^2-2ab+48;ab\le1\), đỉnh \(I\left(\dfrac{1}{9};\dfrac{431}{9}\right)\)

Hàm đồng biến trên khoảng \(\left[\dfrac{1}{9};1\right]\backslash\left\{\dfrac{1}{9}\right\}\)

 \(\Rightarrow f\left(ab\right)_{max}=55\Leftrightarrow ab=1\)

\(\Rightarrow E_{max}=55\Leftrightarrow a=b=1\)

Vậy...

2,

\(ab\le\dfrac{1}{4}\left(a+b\right)^2=1\Rightarrow0\le ab\le1\)

\(E=9a^2b^2+6\left(a^3+b^3\right)+5ab\left(a+b\right)+24ab\)

\(=9a^2b^2+6\left(a+b\right)^3-18ab\left(a+b\right)+5ab\left(a+b\right)+24ab\)

\(=9a^2b^2-2ab+48\)

Đặt \(ab=x\Rightarrow0\le x\le1\)

\(E=9x^2-2x+48=\left(x-1\right)\left(9x+7\right)+55\le55\)

\(E_{max}=55\) khi \(x=1\) hay \(a=b=1\)

Có; \(2a^2+2b^2=5ab\)

\(\Leftrightarrow\left(2a^2-4ab\right)+\left(2b^2-ab\right)=0\)

\(\Leftrightarrow2a\left(a-2b\right)+b\left(2b-a\right)=0\)

\(\Leftrightarrow\left(a-2b\right)\left(2a-b\right)=0\)

\(\Leftrightarrow\left[\begin{array}{nghiempt}a-2b=0\\2a-b=0\end{array}\right.\)\(\Leftrightarrow\left[\begin{array}{nghiempt}a=2b\left(loai\right)\\2a=b\left(tm\right)\end{array}\right.\)

Với: \(2a=b\), ta có: \(P=\frac{a+2a}{a-2a}=\frac{3a}{-a}=-3\)

a)Ta có: ab+ac+bc=-7                        (ab+ac+bc)^2=49

nên

(ab)^2+(bc)^2+(ac)^2=49

nên a^4+b^4+c^4=(a^2+b^2+c^2)^2−2(ab)^2−2(ac)^2−2(bc^)2=98

b) (x^2+y^2+z^2)/(a^2+b^2+c^2)= 
=x^2/a^2+y^2/b^2+z^2/c^2 <=> 
x^2+y^2+z^2=x^2+(a^2/b^2)y^2+ 
+(a^2/c^2)z^2+(b^2/a^2)x^2+y^2+ 
+(b^2/c^2)z^2+(c^2/a^2)x^2+ 
+(c^2/b^2)y^2+z^2 <=> 
[(b^2+c^2)/a^2]x^2+[(a^2+c^2)/b^2]y^2+ 
+[(a^2+b^2)/c^2]z^2 = 0 (*) 
Đặt A=[(b^2+c^2)/a^2]x^2; B=[(a^2+c^2)/b^2]y^2; 
và C=[(a^2+b^2)/c^2]z^2 
Vì a,b,c khác 0 nên suy ra A,B,C đều không âm 
Từ (*) ta có A+B+C=0 
Tổng 3 số không âm bằng 0 thì cả 3 số đều phải bằng 0,tức A=B=C=0 
Vì a,b,c khác 0 nên [(b^2+c^2)/c^2]>0 =>x^2=0 =>x=0 
Tương tự B=C=0 =>y^2=z^2=0 => y=z=0 
Vậy x^2011+y^2011+z^2011=0 
Và x^2008+y^2008+z^2008=0.

Bạn tham khảo lời giải bài 4 link sau:

Câu hỏi của Bonking - Toán lớp 9 | Học trực tuyến

\(\left\{{}\begin{matrix}a;b;c\ge0\\a+b+c=1\end{matrix}\right.\) \(\Rightarrow0\le a;b;c\le1\)

\(\Rightarrow a\left(a-1\right)\le0\Rightarrow a^2\le a\)

\(\Rightarrow\sqrt{2a^2+3a+4}=\sqrt{a^2+a^2+3a+4}\le\sqrt{a^2+a+3a+4}=a+2\)

Tương tự và cộng lại:

\(\Rightarrow M\le a+2+b+2+c+2=7\)

\(M_{max}=7\) khi \(\left(a;b;c\right)=\left(0;0;1\right)\) và các hoán vị