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\(yz\left(y+z\right)+zx\left(z-x\right)-xy\left(x+y\right)\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left[\left(y+z\right)-\left(z-x\right)\right]\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left(y+z\right)+xy\left(z-x\right)\)
\(=y\left(y+z\right)\left(z-x\right)+x\left(z-x\right)\left(z-y\right)\)
\(=\left(z-x\right)\left(yz-xy+xz-xy\right)\)
5(x+y)2+3(x-y)2=8x2+4xy+8y2=4(2x2+xy+2z2)>=5(x+y)2
=> \(\sqrt{2x^2+xy+2y^2}\ge\sqrt{\frac{5\left(x+y\right)^2}{4}}\)= \(\frac{\sqrt{5}\left(x+y\right)}{2}\)
Tương tự. Cộng lại là ra nha. Dấu = xảy ra <=> x=y=z=1/3
b: 5x^2+5y^2+8xy-2x+2y+2=0
=>4x^2+8xy+4y^2+x^2-2x+1+y^2+2y+1=0
=>(x-1)^2+(y+1)^2+(2x+2y)^2=0
=>x=1 và y=-1
M=(1-1)^2015+(1-2)^2016+(-1+1)^2017=1
x+y+z=0
=>x+y=-z; x+z=-y; y+z=-x
\(\left(x+y\right)^2=\left(-z\right)^2=z^2\)
=>\(x^2+2xy+y^2=z^2\)
=>\(z^2-xy=x^2+xy+y^2\)
\(4xy-z^2=4xy-(x^2+2xy+y^2)\)
\(=-(x^2-2xy+y^2)=-(x-y)^2\)
\(xy+2z^2=xy+2(x^2+2xy+y^2)\)
\(=2x^2+5xy+2y^2=(x+2y)(2x+y)\)
\(\left(x+z\right)^2=\left(-y\right)^2=y^2\)
=>\(x^2+2xz+z^2=y^2\)
=>\(y^2-xz=x^2+xz+z^2\)
\(4yz-x^2=4yz-\left(y+z\right)^2\)
\(=4yz-\left(y^2+2yz+z^2\right)=-y^2+2yz-z^2=-\left(y-z\right)^2\)
\(yz+2x^2=yz+2\left(y^2+2yz+z^2\right)\)
\(=2y^2+5yz+2z^2=2y^2+4yz+yz+2z^2\)
=2y(y+2z)+z(y+2z)
=(y+2z)(2y+z)
\(\left(y+z\right)^2=\left(-x\right)^2=x^2\)
=>\(y^2+2yz+z^2=x^2\)
=>\(x^2-yz=y^2+yz+z^2\)
\(4xz-y^2\) =4xz-(x+z)^2
=4xz-\(x^2-2xz-z^2\)
\(=-x^2+2xz-z^2=-\left(x-z\right)^2\)
\(xz+2y^2=xz+2\left(x+z\right)^2\)
\(=xz+2x^2+4xz_{}+2z^2=2x^2+5xz+2z^2\)
\(=2x^2+4xz+xz+2z^2\)
=2x(x+2z)+z(x+2z)
=(x+2z)(2x+z)
2x+y=x+x+y=x-z
\(2y + z = y + (y + z) = y - x = -(x - y)\)
\(2z + x = z + (z + x) = z - y = -(y - z)\)
\(x + 2y = (x + y) + y = -z + y = y - z\)
\(y + 2z = (y + z) + z = -x + z = z - x\)
\(z + 2x = (z + x) + x = -y + x = x - y\)
Ta có: \(A = \frac{4xy - z^2}{xy + 2z^2} \cdot \frac{4yz - x^2}{yz + 2x^2} \cdot \frac{4zx - y^2}{xz + 2y^2}\)
\(=\frac{-(x - y)^2}{(x + 2y)(2x + y)}\cdot\frac{-(y - z)^2}{(y + 2z)(2y + z)}\cdot\frac{-(z - x)^2}{(z + 2x)(2z + x)}\)
\(=\frac{-\left(x-y\right)^2\cdot\left(y-z\right)^2\cdot\left(z-x\right)^2}{(y-z)(x-z)(z-x)[-(x-y)](x-y)[-(y-z)]}\)
\(=\frac{-(x - y)^2 (y - z)^2 (z - x)^2}{-(x - y)^2 (y - z)^2 (z - x)^2}=1\)