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1/
\(M=3x^2-4x+3=3\left(x^2-\frac{4}{3}x+1\right)=3\left(x^2-2x\cdot\frac{2}{3}+\frac{4}{9}\right)+\frac{5}{3}=3\left(x-\frac{2}{3}\right)^2+\frac{5}{3}\ge\frac{5}{3}>0\)
\(N=5x^2-10x+2018=5\left(x^2-2x+1\right)+2013=5\left(x-1\right)^2+2013\ge2013>0\)
\(P=x^2+2y^2-2xy+4y+7=\left(x^2-2xy+y^2\right)+\left(y^2+4y+4\right)+3=\left(x-y\right)^2+\left(y+2\right)^2+3\ge3>0\)
2/
\(A=10x-6x^2+7=-6x^2+10x+7=-6\left(x^2-\frac{10}{6}x+\frac{25}{36}\right)-\frac{11}{6}=-6\left(x-\frac{5}{6}\right)^2-\frac{11}{6}\le-\frac{11}{6}< 0\)
\(B=-3x^2+7x+10=-3\left(x^2-\frac{7}{3}x+\frac{49}{36}\right)-\frac{311}{12}=-3\left(x-\frac{7}{6}\right)^2-\frac{311}{12}\le-\frac{311}{12}< 0\)
\(C=2x-2x^2-y^2+2xy-5=\left(2x-x^2-1\right)-\left(x^2-2xy+y^2\right)-4=-\left(x^2-2x+1\right)-\left(x-y\right)^2-4=-\left(x-1\right)^2-\left(x-y\right)^2-4\)\(\le-4< 0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Cho các số x khác 2y thỏa mãn x2- 2xy - 2y2 - 3x +6y=0
Tính giá trị biểu thức A= x2+ 2xy _y2 - 2x- 2y
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\(4x^2-8x+5=\left(2x\right)^2-2.2.2x+4+1=\left(2x-1\right)^2+1>0\)(luon duong)
\(4x^2-8x+5\)
\(=\left(2x\right)^2-2×2×2x+1+4\)
\(=\left(2x-1\right)^2+1\)
\(\Rightarrow\left(2x-1\right)^2+1>0\)
Vậy biểu thức trên luôn dương !!!
![](https://rs.olm.vn/images/avt/0.png?1311)
\(B=x^2-2x+y^2+4y+6=\left(x^2-2x+1\right)+\left(y^2+4y+4\right)+1=\left(x-1\right)^2+\left(y+2\right)^2+1\ge1>0\forall x,y\)
\(B=x^2-2x+y^2+4y+6\)
\(=x^2-2x+1+y^2+4y+4+1\)
\(=\left(x-1\right)^2+\left(y+2\right)^2+1\ge1>0\forall x,y\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(E=2x^2+y^2-2xy-6x+12=\left(x-y\right)^2+\left(x-3\right)^2+3\ge3>0\)
\(A=x^2+2y^2+2xy+2x+5y+2018\\ =\left(x^2+2xy+y^2\right)+2\left(x+y\right)+1+\left(y^2+3y+2,25\right)+2014,75\\ =\left(x+y\right)^2+2\left(x+y\right)+1+\left(y+1,5\right)^2+2014,75\\ =\left(x+y+1\right)^2+\left(y+1,5\right)^2+2014,75\)
Với mọi x ;y thì \(\left(x+y+1\right)^2\ge1\\ \left(y+1,5\right)^2\ge0\\ \Rightarrow\left(x+y+1\right)^2+\left(y+1,5\right)^2+2014,75\ge2015,75\)
Hay \(A\ge2015,75\) với mọi x;y
Để A=2014,75 thì
\(\left\{{}\begin{matrix}\left(x+y+1\right)^2=0\\\\\left(y+1,5\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+y=-1\\\\y=-1,5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-2,5\\\\y=-1,5\end{matrix}\right.\)
Vậy...
Ko có gìHuyền Anh Kute