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\(Q=x^2-y^2-z^2-2yz-20x\)
\(Q=x^2-\left(y^2+z^2+2yz\right)-20x\)
\(Q=x^2-\left(y+z\right)^2-20x\)
Ta có :
x + y + z = 10
=> y + z = 10 - x
\(Q=x^2-\left(10-x\right)^2-20x\)
\(Q=x^2-\left(100-20x+x^2\right)-20x\)
\(Q=x^2-100+20x-x^2-20x\)
\(Q=-100\)
Do x+y+z=10 nên y+z=10-x, ta có:
\(x^2-20x-y^2-2yz-z^2\) nên bằng \(x^2-20x\left(y^2+2yz+z^2\right)\)
\(=x^2-20x-\left(y+z\right)^2\) <=> = \(x^2-20x-\left(10-x\right)^2\)
=\(x^2-20x-100+20x-x^2\)
và bằng -100 .... tck nha bạn
\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)
\(\Rightarrow xy+yz+xz=0\)
\(\Rightarrow\left\{{}\begin{matrix}xy=-yz--xz\\yz=-xy-xz\\xz=-xy-xz\end{matrix}\right.\)
\(\dfrac{yz}{x^2+2yz}=\dfrac{yz}{x^2+yz-xy-xz}=\dfrac{yz}{\left(x-y\right)\left(x-z\right)}\)
CMTT:
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{xz}{y^2+2xz}=\dfrac{xz}{\left(x-y\right)\left(x-z\right)}\\\dfrac{xy}{z^2+2xy}=\dfrac{xy}{\left(x-y\right)\left(x-z\right)}\\\dfrac{yz}{x^2+2yz}=\dfrac{yz}{\left(x-y\right)\left(x-z\right)}\end{matrix}\right.\)
A=\(\dfrac{xz}{\left(x-y\right)\left(x-z\right)}+\dfrac{xy}{\left(x-y\right)\left(x-z\right)}+\dfrac{yz}{\left(x-y\right)\left(x-z\right)}\)
\(A=\dfrac{xz+xy+yz}{\left(x-y\right)\left(x-z\right)}\left(1\right)\)
mà \(xy+yz+xz=0\)
Từ \(\Rightarrow\dfrac{xz+xy+yz}{\left(x-y\right)\left(x-z\right)}=0\)
Vậy A=0
\(Q=x^2-y^2-z^2-2yz-20x\)
=> \(Q=x^2-y^2-z^2-2yz-20x+100-100\)
=> \(Q=\left(x^2-20x+100\right)-\left(y^2+2yz+z^2\right)-100\)
=> \(Q=\left(x-10\right)^2-\left(y+z\right)^2-10^2\)
=> \(Q=\left(x-10-y-z\right)\left(x-10+y+z\right)-10^2\)
=> \(Q=\left[x-\left(x+y+z\right)-y-z\right]\left[x-\left(x+y+z\right)+y+z\right]-10^2\)
=> \(Q=\left[x-\left(x+y+z\right)-y-z\right]\left[x-\left(x+y+z\right)+y+z\right]-10^2\)
=> \(Q=(-2y-2z).0-10^2\)
=> \(Q=0-10^2\)
=> \(Q=-100\)
TA có :
\(x+y+z=10\Rightarrow\left(x+y+z\right)^2=100\Leftrightarrow x^2+y^2+z^2+2xy+2yz+2xz\)\(Q=x^2-y^2-z^2-2yz-20x\)
\(\Rightarrow100+Q=x^2+y^2+z^2+2xy+2yz+2xz+x^2-y^2-z^2-2yz-20x\)\(=2x^2+2xy+2xz-20x\)
\(=2x\left(x+y+z-10\right)\)
\(=2x\left(x+y+z-x-y-z\right)\)
\(\Rightarrow100+Q=0\Rightarrow Q=-100\)