Bài 2:

a: ĐKXĐ: \(x\notin\left\{0;-1\right\}\)

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

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

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

b: ĐKXĐ: \(x\notin\left\{-23;1\right\}\)

\(\dfrac{2x}{x+23}\cdot\dfrac{3x}{x-1}+\dfrac{2x}{x+23}\cdot\dfrac{23-2x}{x-1}\)

\(=\dfrac{2x}{x+23}\cdot\left(\dfrac{3x}{x-1}+\dfrac{23-2x}{x-1}\right)\)

\(=\dfrac{2x}{x+23}\cdot\dfrac{3x+23-2x}{x-1}\)

\(=\dfrac{2x}{x+23}\cdot\dfrac{x+23}{x-1}=\dfrac{2x}{x-1}\)

Bài 3:

a: Sửa đề: AMCN

Ta có: ABCD là hình bình hành

=>BC=AD(1)

Ta có: M là trung điểm của BC

=>\(BM=MC=\dfrac{BC}{2}\left(2\right)\)

Ta có: N là trung điểm của AD

=>\(NA=ND=\dfrac{AD}{2}\left(3\right)\)

Từ (1),(2),(3) suy ra BM=MC=NA=ND

Xét tứ giác AMCN có

MC//AN

MC=AN

Do đó: AMCN là hình bình hành

b: Xét tứ giác ABMN có

BM//AN

BM=AN

Do đó: ABMN là hình bình hành

Hình bình hành ABMN có \(AB=BM\left(=\dfrac{BC}{2}\right)\)

nên ABMN là hình thoi

c: Ta có: BM//AD

=>\(\widehat{EBM}=\widehat{EAD}\)(hai góc đồng vị)

=>\(\widehat{EBM}=60^0\)

Xét ΔBEM có BE=BM(=BA) và \(\widehat{EBM}=60^0\)

nên ΔBEM đều

=>\(\widehat{BEM}=60^0\)

Xét hình thang ANME có \(\widehat{MEA}=\widehat{EAN}=60^0\)

nên ANME là hình thang cân

=>AM=NE

a) Ta có: \(\left(\dfrac{1}{x^2+x}-\dfrac{2-x}{x+1}\right):\left(\dfrac{1}{x}+x-2\right)\)

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

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

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

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

b) Ta có: \(\left(\dfrac{3x}{1-3x}+\dfrac{2x}{3x+1}\right):\dfrac{6x^2+10x}{1-6x+9x^2}\)

\(=\dfrac{3x\left(3x+1\right)+2x\left(1-3x\right)}{\left(1-3x\right)\left(1+3x\right)}:\dfrac{2x\left(3x+5\right)}{\left(1-3x\right)^2}\)

\(=\dfrac{9x^2+3x+2x-6x^2}{\left(1-3x\right)\left(1+3x\right)}:\dfrac{2x\left(3x+5\right)}{\left(1-3x\right)^2}\)

\(=\dfrac{3x^2+5x}{\left(1-3x\right)\left(1+3x\right)}\cdot\dfrac{\left(1-3x\right)^2}{2x\left(3x+5\right)}\)

\(=\dfrac{x\left(3x+5\right)}{1+3x}\cdot\dfrac{1-3x}{2x\left(3x+5\right)}\)

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

c) Ta có: \(\left(\dfrac{9}{x^3-9x}+\dfrac{1}{x+3}\right):\left(\dfrac{x-3}{x^2+3x}-\dfrac{x}{3x+9}\right)\)

\(=\left(\dfrac{9}{x\left(x-3\right)\left(x+3\right)}+\dfrac{1}{x+3}\right):\left(\dfrac{x-3}{x\left(x+3\right)}-\dfrac{x}{3\left(x+3\right)}\right)\)

\(=\dfrac{9+x\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}:\dfrac{3\left(x-3\right)-x^2}{3x\left(x+3\right)}\)

\(=\dfrac{9+x^2-3x}{x\left(x-3\right)\left(x+3\right)}\cdot\dfrac{3x\left(x+3\right)}{3x-9-x^2}\)

\(=\dfrac{x^2-3x+9}{x-3}\cdot\dfrac{3}{-\left(x^2-3x+9\right)}\)

\(=\dfrac{-3}{x-3}\)

a) (2x+12x−1−2x−12x+1):4x10x−5=(2x+1)2−(2x−1)2(2x−1)(2x+1).10x+54x(2x+12x−1−2x−12x+1):4x10x−5=(2x+1)2−(2x−1)2(2x−1)(2x+1).10x+54x

=4x2+4x+1−4x2+4x−1(2x−1)(2x+1).5(2x+1)4x4x2+4x+1−4x2+4x−1(2x−1)(2x+1).5(2x+1)4x

=8x.5(2x+1)(2x−1)(2

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

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

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

a.

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

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

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

b.

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

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

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

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

\(=\dfrac{2y}{x-y}\)

c.

\(\dfrac{x+5}{2x-4}.\dfrac{4-2x}{x+2}=\dfrac{x+5}{2x-4}.\dfrac{-\left(2x-4\right)}{x+2}=-\dfrac{x+5}{x+2}\)

d.

\(\dfrac{8}{x^2+2x-3}+\dfrac{2}{x+3}+\dfrac{1}{x-1}=\dfrac{8}{\left(x-1\right)\left(x+3\right)}+\dfrac{2\left(x-1\right)}{\left(x-1\right)\left(x+3\right)}+\dfrac{x+3}{\left(x-1\right)\left(x+3\right)}\)

\(=\dfrac{8+2\left(x-1\right)+x+3}{\left(x-1\right)\left(x+3\right)}=\dfrac{3x+9}{\left(x-1\right)\left(x+3\right)}\)

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

a) Ta có: \(\dfrac{1-x}{x^2-2x+1}+\dfrac{x+1}{x-1}\)

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

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

\(=\dfrac{1-x-1}{1-x}=\dfrac{-x}{1-x}=\dfrac{x}{x-1}\)

b) Ta có: \(\dfrac{2x}{3y^4z}\cdot\left(-\dfrac{4y^2z}{5x}\right)\cdot\left(-\dfrac{15y^3}{8xz}\right)\)

\(=\dfrac{2x\cdot4y^2z\cdot15y^3}{3y^4z\cdot5x\cdot8xz}\)

\(=\dfrac{120xy^5z}{120x^2y^4z^2}=\dfrac{y}{xz}\)

a: \(=2x+x^3-5x^4\)

b: \(=\dfrac{8x^2+4x-7x-3}{\left(2x-1\right)\left(2x+1\right)}=\dfrac{8x^2-3x-3}{\left(2x-1\right)\left(2x+1\right)}\)