D = \(-\dfrac{5}{x^2-4x+7}\)

Vì: x² - 4x + 7

= x² - 4x + 4 + 3

= (x - 2)² + 3 \(\ge\) 3 \(\forall\)x

\(\Rightarrow\) \(\dfrac{5}{\left(x-2\right)^2+3}\) \(\le\) \(\dfrac{5}{3}\) \(\forall\)x

\(\Rightarrow\) \(-\dfrac{5}{\left(x-2\right)^2+3}\)\(\ge\)-\(\dfrac{5}{3}\) \(\forall\)x

Dấu"=" xảy ra khi:

x - 2 = 0

\(\Rightarrow\) x = 2

Vậy.............

E = \(\dfrac{2x^2+4x+4}{x^2+2x+4}\)

Ta có:

\(\dfrac{2x^2+4x+4}{x^2+2x+4}\)

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

= 2 - \(\dfrac{4}{x^2+2x+4}\)

Vì:

x² + 2x + 4

= x² + 2x + 1 + 3

= (x + 1)² + 3 \(\ge\) 3 \(\forall\)x

\(\Rightarrow\) \(\dfrac{4}{\left(x+1\right)^2+3}\) \(\le\) \(\dfrac{4}{3}\) \(\forall\)x

\(\Rightarrow\) 2 - \(\dfrac{4}{\left(x+1\right)^2+3}\) \(\le\) \(\dfrac{2}{3}\) \(\forall\)x

Dấu "=" xảy ra khi:

x + 1 = 0

\(\Rightarrow\) x = -1

Vậy...............

F = \(\dfrac{6x+8}{x^2+1}\)

= \(\dfrac{x^2+6x+9-x^2-1}{x^2+1}\)

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

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

Dấu "=" xảy ra khi:

(x + 3)² = 0

\(\Rightarrow\) x + 3 = 0

\(\Rightarrow\) x = -3

Vậy.....................

Bài 5:

a) \(A=x^2-4x+9=\left(x^2-4x+4\right)+5=\left(x-2\right)^2+5\ge5\)

\(minA=5\Leftrightarrow x=2\)

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

\(minB=\dfrac{3}{4}\Leftrightarrow x=\dfrac{1}{2}\)

c) \(C=2x^2-6x=2\left(x^2-3x+\dfrac{9}{4}\right)-\dfrac{9}{2}=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\ge-\dfrac{9}{2}\)

\(minC=-\dfrac{9}{2}\Leftrightarrow x=\dfrac{3}{2}\)

Bài 4:

a) \(M=4x-x^2+3=-\left(x^2-4x+4\right)+7=-\left(x-2\right)^2+7\le7\)

\(maxM=7\Leftrightarrow x=2\)

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

\(maxN=\dfrac{1}{4}\Leftrightarrow x=\dfrac{1}{2}\)

c) \(P=2x-2x^2-5=-2\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{9}{2}=-2\left(x-\dfrac{1}{2}\right)^2-\dfrac{9}{2}\le-\dfrac{9}{2}\)

\(maxP=-\dfrac{9}{2}\Leftrightarrow x=\dfrac{1}{2}\)

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

Vậy \(A_{Min}=-3khix=2\)

Bài \(3\)

\(A=\left(x-5\right)\left(2x+3\right)-2x\left(x-3\right)+x+7\)

\(=2x^2+3x-10x-15-\left(2x^2-6x\right)+x+7\)

\(=2x^2+3x-10x-15-2x^2+6x+x+7\)

\(=\left(2x^2-2x^2\right)+\left(3x-10x+6x+x\right)+\left(-15+7\right)\)

\(=-8\)

Vậy biểu thức không phụ thuộc vào biến

\(B=4\left(y-6\right)-y^2\left(2+3y\right)+y\left(5y-4\right)+3y^2\)

Đề như này à?

Bài \(4\)

\(a,4a^2-16b^2=4\left(a^2-4b^2\right)=4\left(a-2b\right)\left(a+2b\right)\)

\(b,4x^2-4x+1=\left(2x\right)^2-2.2x.1+1^2=\left(2x+1\right)^2\)

\(c,\) ?

\(d,\left(x-y\right)^2-\left(2x-y\right)^2\\ =\left[\left(x-y\right)-\left(2x-y\right)\right]\left[\left(x-y\right)+\left(2x-y\right)\right]\\ =\left(x-y-2x+y\right)\left(x-y+2x-y\right)\\ =\left(-x\right)\left(3x-2y\right)\)

\(e,8x^3-y^3=\left(2x\right)^3-y^3\\ =\left(2x-y\right)\left(4x^2+2xy+y^2\right)\)

\(i,3x+6y+\left(x+2y\right)\\ =3\left(x+2y\right)+\left(x+2y\right)\\ =4\left(x+2y\right)\)

\(j,ax-ay-x+y=\left(ãx-ay\right)-\left(x-y\right)\\ =a\left(x-y\right)-\left(x-y\right)=\left(x-y\right)\left(a-1\right)\)

`k,` `y` hay `y^2` ạ? vì nó mới phân tích được nhân tử.

Tớ xin làm câu k nhé!

\(k)2x^2-y+6x^2y-3y^2\\=(2x^2-y)+(6x^2y-3y^2)\\=(2x^2-y)+3y(2x^2-y)\\=(2x^2-y)(1+3y)\)

#\(Toru\)

\(c)\\1)(2x+y)^2-x^2\\=(2x+y-x)(2x+y+x)\\=(x+y)(3x+y)\\2)?\)

Dấu _ là sao cậu?

#\(Toru\)

Biểu thức nào em?

a) \(A=x^2-2x+5\)

\(A=x^2-2x+1+4\)

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

Có:  \(\left(x-1\right)^2\ge0\Rightarrow\left(x-1\right)^2+4\ge4\)

Dấu '=' xảy ra khi: \(\left(x-1\right)^2=0\Rightarrow x-1=0\Rightarrow x=1\)

Vậy: \(Min_A=4\) tại \(x=1\)

b) \(B=x^2+x+1\)

\(B=x^2+x+\frac{1}{4}+\frac{3}{4}\)

\(B=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\)

Có: \(\left(x+\frac{1}{2}\right)^2\ge0\Rightarrow\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Dấu '=' xảy ra khi: \(\left(x+\frac{1}{2}\right)^2=0\Rightarrow x+\frac{1}{2}=0\Rightarrow x=-\frac{1}{2}\)

Vậy: \(Min_B=\frac{3}{4}\) tại \(x=-\frac{1}{2}\)

c) \(C=4x-x^2+3\)

\(C=-x^2+4x-4+8\) 

\(C=8-\left(x^2-4x+4\right)\)

\(C=8-\left(x-2\right)^2\)

Có: \(\left(x-2\right)^2\ge0\Rightarrow8-\left(x-2\right)^2\le8\)

Dấu '=' xảy ra khi: \(\left(x-2\right)^2=0\Rightarrow x-2=0\Rightarrow x=2\)

Vậy: \(Max_C=8\) tại \(x=2\)