Tìm giá trị lớn nhất của A = -2x2 - 10y2 + 4xy + 4x + 4y + 2013
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\(\frac{2}{3}x-\frac{1}{9}x^2-1\)
\(=-\left(\frac{1}{9}x^2-\frac{2}{3}x+1\right)\)
\(=-\left[\left(\frac{1}{3}x\right)^2-2\cdot\frac{1}{3}x\cdot1+1^2\right]\)
\(=-\left(\frac{1}{3}x-1\right)^2\)
12x3 + 4x2 - 27x - 9
= 4x2 ( 3x + 1 ) - 9 ( 3x + 1 )
= ( 3x +1 ) ( 4x2 -9 )
k nha !
\(12x^3+4x^2-27x-9\)
\(=4x^2\left(3x+1\right)-9\left(3x+1\right)\)
\(=\left(3x+1\right)\left(4x^2-9\right)\)
\(=\left(3x+1\right)\left(2x-3\right)\left(2x+3\right)\)
Chúc bạn học tốt.
\(4x\left(x-1\right)=8\left(x+1\right)\)
\(\Rightarrow4x^2-4x=8x+8\)
\(\Rightarrow4x^2-4x-8x-8=0\)
\(\Rightarrow4x^2-12x-8=0\)
\(\Rightarrow\left(2x\right)^2-2.2x.3+9-17=0\)
\(\Rightarrow\left(2x-3\right)^2=17\)
\(\Rightarrow\orbr{\begin{cases}2x-3=\sqrt{17}\\2x-3=-\sqrt{17}\end{cases}\Rightarrow\orbr{\begin{cases}x=\frac{\sqrt{17}+3}{2}\\x=\frac{-\sqrt{17}+3}{2}\end{cases}}}\)
Chúc bạn học tốt.
Easy \(x^2-n^2-2xy+y^2-m^2+2mn\)
\(=\left(x^2-2xy+y^2\right)-\left(n^2-2mn+m^2\right)\)
\(=\left(x-y\right)^2-\left(n-m\right)^2\)
\(=\left(x-y-n+m\right)\left(x-y+n-m\right)\)
\(x^2-n^2-2xy+y^2-m^2+2mn\)
\(=\left(x^2-2xy+y^2\right)-\left(n^2-2mn+m^2\right)\)
\(=\left(x-y\right)^2-\left(n-m\right)^2\)
\(=\left(x-y-n+m\right)\left(x-y+n-m\right)\)
\(a^2-10a+25-y^2-4yz-4z^2\)
\(=a^2-2.a.5+5^2-y^2-2.y.2z-\left(2z\right)^2\)
\(=\left(a-5\right)^2-\left(y+2z\right)^2\)
\(=\left(a-y-2z-5\right)\left(a+y+2z-5\right)\)
Very easy
\(a^2-10a+25-y^2-4yz-4z^2\)
\(=\left(a-5\right)^2-\left(y+2z\right)^2\)
\(=\left(a-5-y-2z\right)\left(a-5+y+2z\right)\)
\(3a^2c^2+bd+3abc+acd\)
\(=\left(3a^2c^2+3abc\right)+\left(acd+bd\right)\)
\(=3ac\left(ac+b\right)+d\left(ac+b\right)\)
\(=\left(ac+b\right)\left(3ac+d\right)\)
\(-16x^2+8xy-y^2+49\)
\(=49-\left(16x^2-8xy+y^2\right)\)
\(=7^2-\left(4x-y\right)^2\)
\(=\left(7-4x+y\right)\left(7+4x-y\right)\)
\(x^6-x^4+2x^3+2x^2\)
\(=x^2\left(x^4-x^2+2x+2\right)\)
\(=x^2\left[x^4-2x^3+x^2+2x^3-4x^2+2x+2x^2-4x+2\right]\)
\(=x^2\left[x^2\left(x^2-2x+1\right)+2x\left(x^2-2x+1\right)+2\left(x^2-2x+1\right)\right]\)
\(=x^2\left(x^2-2x+1\right)\left(x^2+2x+2\right)\)
\(=x^2\left(x-1\right)^2\left(x^2+2x+2\right)\)
\(A=-2x^2-10y+4xy+4x+4y+2013\)
\(A=-\left(2x^2+10y^2-4xy-4x-4y-2013\right)\)
\(A=-\left(x^2+x^2+y^2+9y^2+2xy-6xy-4x-4y-2013\right)\)
\(A=-\left[\left(x^2+2xy+y^2\right)-4\left(x+y\right)+4+\left(3y\right)^2-2\cdot3y\cdot x+x^2-2017\right]\)
\(A=-\left[\left(x+y\right)^2-2\cdot\left(x+y\right)\cdot2+2^2+\left(3y-x\right)^2-2017\right]\)
\(A=-\left[\left(x+y\right)^2+\left(3y-x\right)^2-2017\right]\)
\(A=2017-\left(x+y\right)^2-\left(3y-x\right)^2\)
\(A=2017-\left[\left(x+y\right)^2-\left(3y-x\right)^2\right]\le2017\)
Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}x+y=0\\3y-x=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+y=0\\3y=x\end{cases}}\Leftrightarrow\hept{\begin{cases}3y+y=0\\x+y=0\end{cases}\Leftrightarrow}\hept{\begin{cases}y=0\\x=0\end{cases}}}\)
\(A=-2x^2-10y^2+4xy+4x+4y+2013\)
\(=-2\left(x-y\right)^2+4\left(x-y\right)-2-8y^2+8y-2+2017\)
\(=-2\left[\left(x-y\right)^2-2\left(x-y\right)+1\right]-8\left(y^2-y+\frac{1}{4}\right)+2017\)
\(=-2\left(x-y-1\right)^2-8\left(y-\frac{1}{2}\right)^2+2017\le2017\forall x;y\)
Dấu "=" xảy ra khi: \(\hept{\begin{cases}x-y-1=0\\y-\frac{1}{2}=0\end{cases}\Rightarrow\hept{\begin{cases}x=\frac{3}{2}\\y=\frac{1}{2}\end{cases}}}\)
Vậy GTLN của A là 2017 khi \(x=\frac{3}{2}\)và \(y=\frac{1}{2}\)
Chúc bạn học tốt.