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a) \(-x^2+6x-16=-\left(x^2-6x+9\right)-7=-\left(x-3\right)^2-7< 0\)
b) \(-5x^2+20x-49=-5\left(x^2-4x+4\right)-29=-5\left(x-2\right)^2-29< 0\)
c) \(-1+x-x^2=-\left(x^2-x+1\right)=-\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{3}{4}=-\left(x-\dfrac{1}{2}\right)^2-\dfrac{3}{4}< 0\)
a. \(2x^2-4x+10=x^2-2x+1+x^2-2x+1+8=\left(x-1\right)^2+\left(x-1\right)^2+8=2\left(x-1\right)^2+8\)
Vì \(2\left(x-1\right)^2\ge0\Rightarrow2\left(x-1\right)^2+8\ge8\)
Vậy...
b. \(x^2+x+1=x^2+x+\frac{1}{4}+\frac{3}{4}=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\)
Vì \(\left(x+\frac{1}{2}\right)^2\ge0\Rightarrow\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)
Vậy..
c. \(2x^2-6x+5=x^2-4x+4+x^2-2x+1=\left(x-2\right)^2+\left(x-1\right)^2\)
Vì \(\hept{\begin{cases}\left(x-2\right)^2\ge0\\\left(x-1\right)^2\ge0\end{cases}}\Rightarrow\left(x-2\right)^2+\left(x-1\right)^2\ge0\)
Vậy...
a,\(-\left(x^2-3x+4\right)\)
\(-\left[\left(x-\frac{3}{2}\right)^2+\frac{7}{4}\right]\)
\(\Leftrightarrow-\left(x-\frac{3}{2}\right)^2-\frac{7}{4}\le-\frac{7}{4}\)(luôn âm)
b\(-2\left(x^2-5x+\frac{15}{2}\right)\)
\(-2\left[\left(x-\frac{5}{2}\right)^2+\frac{5}{4}\right]\)
\(-2\left(x-\frac{5}{4}\right)^2-\frac{5}{2}\le-\frac{5}{2}\)(luôn âm)
c,\(-\left[\left(4x^2-4x+1\right)+\left(2y^2-6y+5\right)\right]\)
\(=-\left[\left(2x-1\right)^2+2\left(y^2-3y+\frac{5}{2}\right)\right]\)
\(=-\left[\left(2x-1\right)^2+2\left(y-\frac{3}{2}\right)^2+\frac{1}{4}\right]\)
\(=-\left[\left(2x-1\right)^2+2\left(y-\frac{3}{2}\right)^2\right]-\frac{1}{4}\le-\frac{1}{4}\)(luôn âm)
a) Đặt \(A=x^2+4x+7\)
\(A=\left(x^2+4x+4\right)+3\)
\(A=\left(x+2\right)^2+3\)
Mà \(\left(x+2\right)^2\ge0\forall x\)
\(\Rightarrow A\ge3>0\)
b) Đặt \(B=4x^2-4x+5\)
\(B=\left(4x^2-4x+1\right)+4\)
\(B=\left(2x-1\right)^2+4\)
Mà \(\left(2x-1\right)^2\ge0\forall x\)
\(\Rightarrow B\ge4>0\)
c) Đặt \(C=x^2+2y^2+2xy-2y+3\)
\(C=\left(x^2+2xy+y^2\right)+\left(y^2-2y+1\right)+2\)
\(C=\left(x+y\right)^2+\left(y-1\right)^2+2\)
Mà \(\left(x+y\right)^2\ge0\forall x;y\)
\(\left(y-1\right)^2\ge0\forall y\)
\(\Rightarrow C\ge2>0\)
\(A=\left(x-y\right)^2\left(z^2-2z+1\right)-2\left(z-1\right)\left(x-y\right)^2+\left(x-y\right)^2\)
\(A=\left(x-y\right)^2\left(z-1\right)^2-2\left(x-y\right)\left(z-1\right)\left(x-y\right)+\left(x-y\right)^2\)
\(A=\left[\left(x-y\right)\left(z-1\right)-\left(x-y\right)\right]^2\ge0\) \(\forall x,y,z\)
\(-5-\left(x-1\right)\left(x+2\right)=-5-\left(x^2+x-2\right)=-5-x^2-x+2\)
\(=-x^2-x-3=-\left(x+\frac{1}{2}\right)^2-\frac{11}{4}< 0,\forall x\inℝ\)
\(a,-x^2+6x-16\)
\(=-x^2+3x+3x-9-5\)
\(=-x\left(x-3\right)+3\left(x-3\right)-5\)
\(=\left(3-x\right)\left(x-3\right)-5\)
\(=-\left(x-3\right)^2-5\le-5\)=>Luôn âm
\(c,-1+x-x^2\)
\(=-x^2+x-1\)
\(=-\left(x^2-x+\frac{1}{2}+\frac{1}{2}\right)\)
\(=-\left(x-\frac{1}{2}\right)^2-\frac{1}{2}\le\frac{-1}{2}\)=>Luôn âm