chung minh rang
x^2 + y^2 +z^2 + 2xy + 2yz + 2zx = ( x+y+z)^2
Những câu hỏi liên quan
Chứng minh đẳng thức:
a, ( x - y - z )2 = x2 + y2 + z2 - 2xy + 2yz - 2zx
b, ( x + y - z )2 = x2 + y2 + z2 + 2xy - 2yz - 2zx
a.\(\left(x^2-y^2-z^2\right)=\left(x-y\right)^2-2z\left(x-y\right)+z^2=x^2-2xy+y^2-2zx+2zy+z^2\)
b.\(\left(x+y-z\right)^2=\left(x+y\right)^2-2z\left(x+y\right)+z^2=x^2+2xy+y^2-2zy-2zx+z^2\)
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Chứng minh: \(\left(x+y+z\right)^2=x^2+y^2+z^2+2xy+2yz+2zx\)
=[(x+y)+z]2
=(x+y)2+2(x+y)z+z2
=x2+2xy+y2+2xz+2yz+z2
=x2+y2+z2+2xy+2yz+2xz
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Chứng minh đẳng thức
a, (x-y-z)^2=x^2 + y^2+z^2-2xy+2yz-2zx
b, ( x+y-z)^2=x^2+y^2+z^2+2xy-2yz-2zx
c, ( x-y)(x^3+x^2y+xy^2+y^3)=5x(x+1)
d, ( x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)=x^5+y^5
Giúp mk vs ạ mk đang cần
a, b, nhân vào là ra à
c, nghe cứ là lạ
d, cũng nhân là ra hà
\(=x^5-x^4y+x^3y^2-x^2y^3+xy^4+x^4y-x^3y^2+x^2y^3-xy^4+y^5=x^5+y^5\)
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a) Ta có: \(VT=\left(x-y-z\right)^2\)
\(=\left(x-y-z\right)\left(x-y-z\right)\)
\(=x^2-xy-xz-yx+y^2+yz-zx+zy+z^2\)
\(=x^2+y^2+z^2-2xy+2yz-2xz\)
=VP(đpcm)
b) Ta có: \(VT=\left(x+y-z\right)^2\)
\(=\left(x+y-z\right)\left(x+y-z\right)\)
\(=x^2+xy-xz+yx+y^2-yz-zx-zy+z^2\)
\(=x^2+y^2+z^2+2xy-2yz-2zx\)
=VP(đpcm)
c) Sửa đề: Chứng minh \(\left(x-y\right)\left(x^3+x^2y+xy^2+y^3\right)=x^4-y^4\)
Ta có: \(VT=\left(x-y\right)\left(x^3+x^2y+xy^2+y^3\right)\)
\(=x^4+x^3y+x^2y^2+xy^3-x^3y-x^2y^2-xy^3-y^4\)
\(=x^4-y^4\)
=VP(đpcm)
d) Ta có: \(VT=\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)\)
\(=x^5-x^4y+x^3y^2-x^2y^3+xy^4+x^4y-x^3y^2+x^2y^3-xy^4+y^5\)
\(=x^5+y^5\)
=VP(đpcm)
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cho x,y,z duong va x+y+z=1. chung minh rang 1/x^2+2yz+1/y^2+2xz+1/z^2+2xy>=9
thank, giup minh mik cho 3 tick
\(\frac{1}{x^2+2yz}+\frac{1}{y^2+2xz}+\frac{1}{z^2+2xy}\ge\frac{\left(1+1+1\right)^2}{x^2+y^2+z^2+2xy+2yz+2xz}=\frac{9}{\left(x+y+z\right)^2}=9\)
Dấu "=" xảy ra <=> \(\hept{\begin{cases}x+y+z=1\\x=y=z\end{cases}\Leftrightarrow x=y=z=\frac{1}{3}}\)
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Xem thêm câu trả lời
rút gọn: x^2+^y2+z^2-2xy-2zx-2yz/x^2-2xy-y^2-z^2
x2 +y2 +z2 -2xy-2zx-2yz=(x-y-z)2 -4yz=(x-y-z)2 - \(2.\sqrt{yz^2}\)=\(\left(x-y-z-2\sqrt{yz}\right)+\left(x-y-z+2\sqrt{yz}\right)\)
x2 -2xy - y2 -z2 =(x-y)2 -z2 =(x-y-z)(x-y+z)
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Chứng minh rằng: (x+y-z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz +2zx
\(\left(x+y+z\right)^2\)
\(=\left(x+y\right)^2+2\left(x+y\right)z+z^2\)
\(=x^2+2xy+y^2+2xz+2xy+z^2\)
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CMR
(x-y-z)^2 = x^2 + y^2 + z^2 - 2xy + 2yz - 2zx
\(\left(x-y-z\right)^2=\left[\left(x-y\right)-z\right]^2\)
\(=\left(x-y\right)^2-2z\left(x-y\right)+z^2\)
\(=x^2-2xy+y^2-2xz+2yz+z^2\)
\(=x^2+y^2+z^2-2xy+2yz-2xz\)\(\left(đpcm\right)\)
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Áp dụng HĐT (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca đó bạn.
Ta có: (x - y + z)^2 >= 0
<=> x^2 + y^2 + z^2 - 2xy + 2xz - 2yz >= 0
<=> x^2 + y^2 + z^2 >= 2xy - 2xz + 2yz
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Xem thêm câu trả lời
\(\frac{x^2+y^2+z^2-2xy-2yz+2zx}{x^2-2xy+y^2-z^2}\)
\(\frac{x^2+y^2+z^2-2xy-2yz+2zx}{x^2-2xy+y^2-z^2}=\frac{\left(x-y+z\right)^2}{\left(x-y\right)^2-z^2}=\frac{\left(x-y+z\right)^2}{\left(x-y-z\right)\left(x-y+z\right)}=\frac{x-y+z}{x-y-z}\)
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Chung minh dang thuc
a)(x+a)(x+b)=x2+(a+b)x+ab
b)(x+a)(x+b)(x+c)=x3+(a+b+c)x2+(ab+bc+ca)x+abc
c)(x-y-z)2=x2+y2+z2-2xy+2yz-2zx
d)(x+y-z)2=x2+y2+z2+2xy-2yz-2zx
e)(x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x)
lam on giup minh nha minh can rat gap thanks