\(\left(x-y+z\right)^2+\left(z-y\right)^2+2.\left(x-y+z\right).\left(y-z\right)\)
Những câu hỏi liên quan
ta có : x^2+1x^2+xy+yz+zxxleft(x+yright)+zleft(x+yright)left(x+yright)left(x+zright)Tương tự ta đc y^2+1left(y+xright)left(y+zright) z^2+1left(z+xright)left(z+yright)ĐẶt Axsqrt{frac{left(1+y^2right)left(1+z^2right)}{left(1+x^2right)}}+ysqrt{frac{left(1+z^2right)left(1+x^2right)}{left(1+y^2right)}}+zsqrt{frac{left(1+x^2right)left(1+y^2right)}{left(1+z^2right)}}Rightarrow Axsqrt{frac{left(x+yright)left(y+zright)left(z+xright)left(y+zright)}{left(x+yright)left(x+zright)}}+ysq...
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ta có : \(x^2+1=x^2+xy+yz+zx=x\left(x+y\right)+z\left(x+y\right)=\left(x+y\right)\left(x+z\right)\)
Tương tự ta đc \(y^2+1=\left(y+x\right)\left(y+z\right)\)
\(z^2+1=\left(z+x\right)\left(z+y\right)\)
ĐẶt \(A=x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{\left(1+x^2\right)}}+y\sqrt{\frac{\left(1+z^2\right)\left(1+x^2\right)}{\left(1+y^2\right)}}+z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{\left(1+z^2\right)}}\)
\(\Rightarrow A=x\sqrt{\frac{\left(x+y\right)\left(y+z\right)\left(z+x\right)\left(y+z\right)}{\left(x+y\right)\left(x+z\right)}}+y\sqrt{\frac{\left(z+x\right)\left(z+y\right)\left(x+y\right)\left(x+z\right)}{\left(x+y\right)\left(y+z\right)}}+z\sqrt{\frac{\left(x+y\right)\left(x+z\right)\left(y+z\right)\left(y+x\right)}{\left(z+x\right)\left(z+y\right)}}\)
\(\Rightarrow A=x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)=2\left(xy+yz+zx\right)=2\)
thực hiện phép tính
a,x^3+left[frac{xleft(2y^3-x^3right)}{x^3+y^3}right]^3-left[frac{yleft(2x^3-y^3right)}{x^3+y^3}right]^3
b,frac{frac{xleft(x+yright)}{x-y}+frac{xleft(x+zright)}{x-z}}{1+frac{left(y-zright)^2}{left(x-yright)left(x-zright)}}+frac{frac{yleft(y+zright)}{y-z}+frac{yleft(y+xright)}{y-x}}{1+frac{left(z-xright)^2}{left(y-zright)left(y-xright)}}+frac{frac{zleft(z+xright)}{z-x}+frac{zleft(z+yright)}{z-y}}{1+frac{left(x-yright)^2}{left(z-xright)left(z-yright)}}
c,left[frac{y+z-2x}{fra...
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thực hiện phép tính
a,\(x^3+\left[\frac{x\left(2y^3-x^3\right)}{x^3+y^3}\right]^3-\left[\frac{y\left(2x^3-y^3\right)}{x^3+y^3}\right]^3\)
b,\(\frac{\frac{x\left(x+y\right)}{x-y}+\frac{x\left(x+z\right)}{x-z}}{1+\frac{\left(y-z\right)^2}{\left(x-y\right)\left(x-z\right)}}+\frac{\frac{y\left(y+z\right)}{y-z}+\frac{y\left(y+x\right)}{y-x}}{1+\frac{\left(z-x\right)^2}{\left(y-z\right)\left(y-x\right)}}+\frac{\frac{z\left(z+x\right)}{z-x}+\frac{z\left(z+y\right)}{z-y}}{1+\frac{\left(x-y\right)^2}{\left(z-x\right)\left(z-y\right)}}\)
c,\(\left[\frac{y+z-2x}{\frac{\left(y-z\right)^3}{y^3-z^3}+\frac{\left(x-y\right)\left(x-z\right)}{y^2+yz+z^2}}+\frac{z+x-2y}{\frac{\left(z-x\right)^3}{z^3-x^3}+\frac{\left(y-z\right)\left(y-x\right)}{z^2+xz+x^2}}+\frac{x+y-2z}{\frac{\left(x-y\right)^3}{x^3-y^3}+\frac{\left(z-x\right)\left(z-y\right)}{x^2+xy+y^2}}\right]:\frac{1}{x+y+z}\)
Tính:
\(\dfrac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2-xz}{\left(y+z\right)\left(y+x\right)}+\dfrac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)
\(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
Tính:
\(\dfrac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2-xz}{\left(y+z\right)\left(y+x\right)}+\dfrac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)
\(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
\(\dfrac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2-xz}{\left(y+z\right)\left(x+y\right)}+\dfrac{z^2-xy}{\left(x+z\right)\left(z+y\right)}\)
\(=\dfrac{\left(x^2-yz\right)\left(y+z\right)+\left(y^2-xz\right)\left(x+z\right)+\left(z^2-xy\right)\left(x+y\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)
\(\left\{{}\begin{matrix}\left(x^2-yz\right)\left(y+z\right)=x^2y+x^2z-y^2z-yz^2\\\left(y^2-xz\right)\left(x+z\right)=y^2x+y^2z-x^2z-xz^2\\\left(z^2-xy\right)\left(x+y\right)=z^2x+z^2y-x^2y-xy^2\end{matrix}\right.\)
Đa thức trên bằng 0
\(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
\(=\dfrac{-x^2}{\left(x-y\right)\left(z-x\right)}+\dfrac{-y^2}{\left(x-y\right)\left(y-z\right)}+\dfrac{-z^2}{\left(z-x\right)\left(y-z\right)}\)
\(=\dfrac{-x^2\left(y-z\right)-y^2\left(z-x\right)-z^2\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
Xét: \(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z+y^2z-xy^2+z^2\left(x-y\right)\)
\(\)\(=xy\left(x-y\right)-z\left(x^2-y^2\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left(xy-xz-yz+z^2\right)\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)
\(=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
Thêm dấu - đằng trc nữa suy ra bt có giá trị bằng 1 :P
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a) dfrac{1}{left(x-yright)left(y-zright)}+dfrac{1}{left(y-zright)left(z-xright)}+dfrac{1}{left(z-xright)left(x-yright)}
b) dfrac{1}{xleft(x-yright)left(x-zright)}+dfrac{1}{yleft(y-zright)left(y-xright)}+dfrac{1}{zleft(z-xright)left(z-yright)}
c) dfrac{x^2}{left(x-yright)left(x-zright)}+dfrac{y^2}{left(y-xright)left(y-zright)}+dfrac{z^2}{left(z-xright)left(z-yright)}
Đọc tiếp
a) \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-x\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\)
b) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-z\right)\left(y-x\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}\)
c) \(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
a: \(=\dfrac{1}{\left(x-y\right)\left(y-z\right)}-\dfrac{1}{\left(y-z\right)\left(x-z\right)}-\dfrac{1}{\left(x-y\right)\left(x-z\right)}\)
\(=\dfrac{x-z-x+y-y+z}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}=0\)
b: \(=\dfrac{1}{x\left(x-y\right)\left(x-z\right)}-\dfrac{1}{y\left(x-y\right)\left(y-z\right)}+\dfrac{1}{z\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{y^2z-yz^2-x^2z+xz^2+xy\left(x-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{z\left(y^2-x^2\right)-z^2\left(y-x\right)-xy\left(y-x\right)}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{\left(x-y\right)\left[-z\left(x+y\right)+z^2+xy\right]}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{-zx-zy+z^2+xy}{xyz\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{z\left(z-x\right)-y\left(z-x\right)}{xyz\left(y-z\right)\left(x-z\right)}=\dfrac{1}{xyz}\)
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Chứng minh đẳng thức:
a) dfrac{y}{left(x-yright)left(y-zright)}+dfrac{z}{left(y-zright)left(z-xright)}+dfrac{x}{left(z-xright)left(x-yright)0}
b) dfrac{x^2}{left(x-yright)left(y-zright)}+dfrac{y^2}{left(y-zright)left(y-xright)}+dfrac{z^2}{left(z-xright)left(z-yright)1}
c) dfrac{1}{xleft(x-yright)left(x-zright)}+dfrac{1}{yleft(y-zright)left(y-xright)}+dfrac{1}{zleft(z-xright)left(z-yright)}dfrac{1}{xyz}
Đọc tiếp
Chứng minh đẳng thức:
a) \(\dfrac{y}{\left(x-y\right)\left(y-z\right)}+\dfrac{z}{\left(y-z\right)\left(z-x\right)}+\dfrac{x}{\left(z-x\right)\left(x-y\right)=0}\)
b) \(\dfrac{x^2}{\left(x-y\right)\left(y-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)=1}\)
c) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-z\right)\left(y-x\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}=\dfrac{1}{xyz}\)
a: \(\dfrac{y}{\left(x-y\right)\left(y-z\right)}-\dfrac{z}{\left(y-z\right)\left(x-z\right)}-\dfrac{x}{\left(x-y\right)\left(x-z\right)}\)
\(=\dfrac{xy-yz-xz+yz-xy+xz}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
=0
c: \(=\dfrac{1}{x\left(x-y\right)\left(x-z\right)}-\dfrac{1}{y\left(y-z\right)\left(x-y\right)}+\dfrac{1}{z\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{zy\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{zy^2-z^2y-x^2z+xz^2+xy\left(x-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\dfrac{1}{xyz}\)
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Cho x , y , z
\(\left(x-y\right)^2\)+\(\left(y-z\right)^2\)+\(\left(z-x\right)^2\)=\(\left(x+y-2z\right)^2\)+\(\left(y+z-2x\right)^2+\left(z+x-2y\right)^2\)
cmr: x=y=z
Phân tích vế trái ta được: 2(x2 + y2 + z2 − (xy + yz + zx)
Phân tích vế phải ta được: 6(x2 + y2 + z2 − (xy + yz + zx)
Vì VT = VP nên VP - VT=0
→ 4(x2 + y2 + z2 − (xy + yz + zx)) = 0
→2(2 (x2 + y2 + z2 − (xy + yz + zx))) = 0
→2((x − y)2 + (y − z)2 + (z − x)2) = 0
→(x − y)2 + (y − z)2 + (z − x)2 = 0
→(x − y)2 = 0; (y − z)2 = 0; (z − x)2 = 0
→x = y = z
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Tính:
a) \(\dfrac{x}{\left(x-y\right)\left(x-z\right)}+\dfrac{y}{\left(y-x\right)\left(y-z\right)}+\dfrac{z}{\left(z-x\right)\left(z-y\right)}\)
b) \(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-x\right)\left(y-z\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
a,
\(-\dfrac{x}{\left(x-y\right)\left(z-x\right)}-\dfrac{y}{\left(x-y\right)\left(y-z\right)}-\dfrac{z}{\left(z-x\right)\left(y-z\right)}\)
\(\dfrac{-x\left(y-z\right)-y\left(z-x\right)-z\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(\dfrac{-xy+xz-yz+xy-zx+yz}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
= 0
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Tính:
a) dfrac{x^2}{left(x-yright)left(x-zright)}+dfrac{y^2}{left(y-zright)left(y-xright)}+dfrac{z^2}{left(z-xright)left(z-yright)}
b) dfrac{x^2-yz}{left(x+yright)left(x+zright)}+dfrac{y^2-zx}{left(y+zright)left(y+xright)}+dfrac{z^2-xy}{left(z+xright)left(z+yright)}
c) dfrac{1}{xleft(x-yright)left(x-zright)}+dfrac{1}{yleft(y-xright)left(y-zright)}+dfrac{1}{zleft(z-xright)left(z-yright)}
d) dfrac{1}{xleft(x+1right)}+dfrac{1}{left(x+1right)left(x+2right)}+dfrac{1}{left(x+2right)left(x+3right)}...
Đọc tiếp
Tính:
a) \(\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}\)
b) \(\dfrac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\dfrac{y^2-zx}{\left(y+z\right)\left(y+x\right)}+\dfrac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)
c) \(\dfrac{1}{x\left(x-y\right)\left(x-z\right)}+\dfrac{1}{y\left(y-x\right)\left(y-z\right)}+\dfrac{1}{z\left(z-x\right)\left(z-y\right)}\)
d) \(\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+...+\dfrac{1}{\left(x+99\right)\left(x+100\right)}\)
Giúp mình với!!! Mình cần gấp!!! 10 giờ sáng mai cần gấp nha !!!
d)
\(\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+.....+\dfrac{1}{\left(x+99\right)\left(x+100\right)}\)=\(\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+\dfrac{1}{x+2}-\dfrac{1}{x+3}+.....-\dfrac{1}{x+99}+\dfrac{1}{x+100}\)=\(\dfrac{1}{x}-\dfrac{1}{x+100}\)
=\(\dfrac{x+100}{x\left(x+100\right)}-\dfrac{x}{x\left(x+100\right)}\)
=\(\dfrac{x+100-x}{x\left(x+100\right)}=\dfrac{100}{x\left(x+100\right)}\)
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cho x,y,z>0 thỏa mãn \(\left(x^2+y^2\right)\left(y^2+z^2\right)\left(z^2+x^2\right)=8\)
Tìm giá trị nhỏ nhất của S=\(xyz\left(x+y+z\right)^3\)
(có thể dùng BDT \(\left(x+y\right)\left(y+z\right)\left(z+x\right)\ge\dfrac{8}{9}\left(x+y+z\right)\left(xy+yz+zx\right)\))
tks mn<3