\(2x^2+2y^2-2xy-4x-4y+8\)

\(=x^2-2xy+y^2+x^2-4x+y^2-4y+8\)

\(=\left(x-y\right)^2+x^2-4x+4+y^2-4x+4\)

\(=\left(x-y\right)^2+\left(x-2\right)^2+\left(y-2\right)^2\ge0\)

\(\RightarrowĐPCM\)

a)\(x^2+4y^2-2x+4y+2\)

\(=\left(x^2-2x+1\right)+\left(4y^2+4y+1\right)\)

\(=\left(x-1\right)^2+\left(2y+1\right)^2\ge0\)(đúng)

b) Sửa đề

\(3y^2+x^2+2xy+2x+6y+3\)

\(=\left(x^2+y^2+2xy\right)+2y^2+2x+6y+3\)

\(=\left(x+y\right)^2+2\left(x+y\right)+1+2y^2+4y+2\)

\(=\left(x+y+1\right)^2+2\left(y+1\right)^2\ge0\) (đúng)

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Bộ bạn ko biết ghi đề bài hả xuống tiểu học đi nha

\(x^4+4x^3+6x^2+4x+1\)

\(=\left(x^4+2x^3+x^2\right)+\left(2x^3+4x^2+2x\right)+\left(x^2+2x+1\right)\)

\(=x^2\left(x^2+2x+1\right)+2x\left(x^2+2x+1\right)+\left(x^2+2x+1\right)\)

\(=\left(x^2+2x+1\right)\left(x^2+2x+1\right)=\left(x+1\right)^4\ge0;\forall x\in R\)

d/ \(x^2+4x-2xy-4y+y^2=\left(x-y\right)^2+4\left(x-y\right)=\left(x-y\right)\left(x-y+4\right)\)

e/ \(x^2+2x+1-16y^2=\left(x+1\right)^2-\left(4y\right)^2=\left(x+1-4y\right)\left(x+1+4y\right)\)

a: Ta có: \(x^2-8x+20\)

\(=x^2-8x+16+4\)

\(=\left(x-4\right)^2+4>0\forall x\)

b: Ta có: \(-x^2+6x-19\)

\(=-\left(x^2-6x+19\right)\)

\(=-\left(x^2-6x+9+10\right)\)

\(=-\left(x-3\right)^2-10< 0\forall x\)

a) đặt \(A=x^2+x+1\)

\(=x^2+2\cdot x\cdot\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2-\dfrac{1}{4}+1\)

\(=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)

Dấu "=' xảy ra khi \(x=-\dfrac{1}{2}\)

Vậy \(MIN_A=\dfrac{3}{4}\) khi \(x=-\dfrac{1}{2}\)

b) đặt \(B=2+x-x^2\)

\(=-x^2+x+2\)

\(=-\left(x^2-x-2\right)\)

\(=-\left[x^2-2\cdot x\cdot\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2-\dfrac{1}{4}-2\right]\)

\(=-\left[\left(x-\dfrac{1}{2}\right)^2-\dfrac{9}{4}\right]\)

\(=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{9}{4}\le\dfrac{9}{4}\)

Dấu "=" xảy ra khi \(x=\dfrac{1}{2}\)

Vậy \(MAX_B=\dfrac{9}{4}\) khi \(x=\dfrac{1}{2}\)

c) đặt \(C=x^2-4x+1\)

\(=x^2-2\cdot x\cdot2+2^2-4+1\)

\(=\left(x-2\right)^2-3\ge-3\)

Dấu "=" xảy ra khi \(x=2\)

Vậy \(MIN_c=-3\) khi \(x=2\)

d) đặt \(D=4x^2+4x+11\)

\(=\left(2x\right)^2+2\cdot2x\cdot1+1^2-1+11\)

\(=\left(2x+1\right)^2+10\ge10\)

Dấu "=" xảy ra khi \(x=-\dfrac{1}{2}\)

Vậy \(MIN_D=10\) khi \(x=-\dfrac{1}{2}\)

mấy câu còn lại tương tự

\(a,3x\left(3x+6\right)=9x^2+18x\)

\(b,-\dfrac{1}{2}xy\left(4x^2+6x\right)\)

\(=-2x^3y-3x^2y\)

\(c,-2x^2y^3\left(\dfrac{1}{2}xy+4y^2\right)\)

\(=-x^3y^4-8x^2y^5\)

\(d,-6x^2\left(\dfrac{1}{3}xy^2-\dfrac{1}{2}y\right)\)

\(=-2x^3y^2+3x^2y\)

#\(Urushi\)

a) 3x(3x+6) = 9x^2 + 18x b) -1/2xy(4x^2+6x) = -2x^3y - 3xy c) -2x^2y^3(1/2xy+4y^2 ) = -x^2y^2 - 8x^2y^5 d) -6x^2(1/3xy^2-1/2y) = -2xy + 3x^2y

Vậy kết quả của các biểu thức là: a) 9x^2 + 18x b) -2x^3y - 3xy c) -x^2y^2 - 8x^2y^5 d) -2xy + 3x^2y