a.

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

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

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

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

2.

\(x^3-4x^3+3x=0\Leftrightarrow x\left(x^2-4x+3\right)=0\)

\(\Leftrightarrow x\left(x-1\right)\left(x-3\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=0\left(loại\right)\\x=1\left(loại\right)\\x=3\end{matrix}\right.\)

Với \(x=3\Rightarrow A=\dfrac{3^2+3.3+1}{3+1}=\dfrac{19}{4}\)

4.linda sometimes brings her home made after the class

Linh 6A3(THCS Mai Đình) à

Bài 4:

a. Vì $\triangle ABC\sim \triangle A'B'C'$ nên:

$\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{AC}{A'C'}(1)$ và $\widehat{ABC}=\widehat{A'B'C'}$

$\frac{DB}{DC}=\frac{D'B'}{D'C}$

$\Rightarrow \frac{BD}{BC}=\frac{D'B'}{B'C'}$

$\Rightarrow \frac{BD}{B'D'}=\frac{BC}{B'C'}(2)$

Từ $(1); (2)\Rightarrow \frac{BD}{B'D'}=\frac{BC}{B'C'}=\frac{AB}{A'B'}$

Xét tam giác $ABD$ và $A'B'D'$ có:

$\widehat{ABD}=\widehat{ABC}=\widehat{A'B'C'}=\widehat{A'B'D'}$

$\frac{AB}{A'B'}=\frac{BD}{B'D'}$

$\Rightarrow \triangle ABD\sim \triangle A'B'D'$ (c.g.c)

b.

Từ tam giác đồng dạng phần a và (1) suy ra:
$\frac{AD}{A'D'}=\frac{AB}{A'B'}=\frac{BC}{B'C'}$

$\Rightarrow AD.B'C'=BC.A'D'$

Hình bài 4:

14:

a: \(\frac{7x-1}{2x^2+6x}=\frac{7x-1}{2x\left(x+3\right)}=\frac{\left(7x-1\right)\left(x-3\right)}{2x\left(x+3\right)\left(x-3\right)}=\frac{7x^2-22x+3}{2x\left(x+3\right)\left(x-3\right)}\)

\(\frac{5-3x}{x^2-9}=\frac{2x\left(5-3x\right)}{2x\left(x-3\right)\left(x+3\right)}=\frac{10x-6x^2}{2x\left(x-3\right)\left(x+3\right)}\)

b: \(\frac{x+1}{x-x^2}=\frac{-\left(x+1\right)}{x^2-x}=\frac{-\left(x+1\right)}{x\left(x-1\right)}=\frac{-\left(x+1\right)\cdot2\left(x-1\right)}{2x\left(x-1\right)^2}=\frac{-2x^2+2}{2x\left(x-1\right)^2}\)

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

c: \(\frac{4x^2-3x+5}{x^3-1}=\frac{4x^2-3x+5}{\left(x-1\right)\cdot\left(x^2+x+1\right)}\)

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

\(\frac{6}{x-1}=\frac{6\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{6x^2+6x+6}{\left(x-1\right)\left(x_{}^2+x+1\right)}\)

d: \(\frac{7}{5x}=\frac{7\cdot2\cdot\left(x-2y\right)\left(x+2y\right)}{5x\cdot2\cdot\left(x-2y\right)\left(x+2y\right)}=\frac{14\left(x^2-4y^2\right)}{10x\left(x-2y\right)\left(x+2y\right)}=\frac{14x^2-56y^2}{10x\left(x-2y\right)\left(x+2y\right)}\)

\(\frac{4}{x-2y}=\frac{4\cdot5x\cdot2\cdot\left(x+2y\right)}{\left(x-2y\right)\cdot5x\cdot2\cdot\left(x+2y\right)}=\frac{40x\left(x+2y\right)}{10x\left(x-2y\right)\left(x+2y\right)}=\frac{40x^2+80xy}{10x\left(x-2y\right)\left(x+2y\right)}\)

\(\frac{y-x}{8y^2-2x^2}=\frac{x-y}{2x^2-8y^2}=\frac{x-y}{2\left(x-2y\right)\left(x+2y\right)}=\frac{5x\left(x-y\right)}{2\cdot5x\left(x-2y\right)\left(x+2y\right)}=\frac{5x^2-5xy}{10x\left(x-2y\right)\left(x+2y\right)}\)

e: \(\frac{5x^2}{x^3+6x^2+12x+8}=\frac{5x^2}{\left(x+2\right)^3}=\frac{5x^2\cdot2}{2\left(x+2\right)^3}=\frac{10x^2}{2\left(x+2\right)^3}\)

\(\frac{4x}{x^2+4x+4}=\frac{4x}{\left(x+2\right)^2}=\frac{4x\cdot2\cdot\left(x+2\right)}{2\left(x+2\right)^3}=\frac{8x^2+16x}{2\left(x+2\right)^3}\)

\(\frac{3}{2x+4}=\frac{3}{2\left(x+2\right)}=\frac{3\left(x+2\right)^2}{2\left(x+2\right)^3}=\frac{3\left(x^2+4x+4\right)}{2\left(x+2\right)^3}=\frac{3x^2+12x+12}{2\left(x+2\right)^3}\)

13:

a: \(\frac{25}{14x^2y}=\frac{25\cdot3\cdot y^4}{14x^2y\cdot3y^4}=\frac{75y^4}{45x^2y^5}\)

\(\frac{14}{21xy^5}=\frac{14\cdot2\cdot x}{2x\cdot21xy^5}=\frac{28x}{42x^2y^5}\)

b: \(\frac{11}{102x^4y}=\frac{11\cdot y^2}{102x^4y\cdot y^2}=\frac{11y^2}{102x^4y^3}\)

\(\frac{3}{34xy^3}=\frac{3\cdot x^3\cdot3}{34xy^3\cdot3x^3}=\frac{9x^3}{102x^4y^3}\)

c: \(\frac{3x+1}{12xy^4}=\frac{\left(3x+1\right)\cdot3\cdot x}{12xy^4\cdot3x}=\frac{9x^2+3x}{36x^2y^4}\)

\(\frac{y-2}{9x^2y^3}=\frac{\left(y-2\right)\cdot4\cdot y}{9x^2y^3\cdot4y}=\frac{4y^2-8y}{36x^2y^4}\)

d: \(\frac{1}{6x^3y^2}=\frac{1\cdot6\cdot xy^2}{6x^3y^2\cdot6xy^2}=\frac{6xy^2}{36x^4y^4}\)

\(\frac{x+1}{9x^2y^4}=\frac{\left(x+1\right)\cdot4\cdot x^2}{9x^2y^4\cdot4x^2}=\frac{4x^3+4x^2}{36x^4y^4}\)

\(\frac{x-1}{4xy^3}=\frac{\left(x-1\right)\cdot9\cdot x^3y}{4xy^3\cdot9x^3y}=\frac{9x^4y-9x^3y}{36x^4y^4}\)

e: \(\frac{3+2x}{10x^4y}=\frac{\left(2x+3\right)\cdot4y^4}{10x^4y\cdot4y^4}=\frac{8xy^4+12y^4}{40x^4y^5}=\frac{3\left(8xy^4+12y^4\right)}{3\cdot40x^4y^4}=\frac{24xy^4+36y^4}{120x^4y^4}\)

\(\frac{5}{8x^2y^2}=\frac{5\cdot5\cdot x^2y^3}{8x^2y^2\cdot5x^2y^3}=\frac{25x^2y^3}{40x^4y^5}=\frac{25x^2y^3\cdot3}{40x^4y^5\cdot3}=\frac{75x^2y^3}{120x^4y^5}\)

\(\frac{2}{3xy^5}=\frac{2\cdot40\cdot x^3}{3xy^5\cdot40x^3}=\frac{80x^3}{120x^4y^5}\)

f: \(\frac{4x-4}{2x\left(x+3\right)}=\frac{2\cdot\left(x-1\right)}{2x\cdot\left(x+3\right)}=\frac{x-1}{x\left(x+3\right)}=\frac{\left(x-1\right)\cdot3\left(x+1\right)}{3x\left(x+3\right)\left(x+1\right)}=\frac{3x^2-3}{3x\left(x+3\right)\left(x+1\right)}\)

\(\frac{x-3}{3x\left(x+1\right)}=\frac{\left(x-3\right)\left(x+3\right)}{3x\left(x+1\right)\left(x+3\right)}=\frac{x^2-9}{3x\left(x+1\right)\left(x+3\right)}\)

g: \(\frac{2x}{\left(x+2\right)^3}=\frac{2x\cdot2x}{2x\left(x+2\right)^3}=\frac{4x^2}{2x\left(x+2\right)^3}\)

\(\frac{x-2}{2x\left(x+2\right)^2}=\frac{\left(x-2\right)\left(x+2\right)}{2x\left(x+2\right)^2\cdot\left(x+2\right)}=\frac{x^2-4}{2x\left(x+2\right)^3}\)

h: \(\frac{5}{3x^3-12x}=\frac{5}{3x\left(x^2-4\right)}=\frac{5}{3x\left(x-2\right)\left(x+2\right)}=\frac{5\cdot2\left(x+3\right)}{3x\left(x-2\right)\left(x+2\right)\cdot2\left(x+3\right)}=\frac{10x+30}{6x\left(x-2\right)\left(x+2\right)\left(x+3\right)}\)

\(\frac{3}{\left(2x+4\right)\left(x+3\right)}=\frac{3}{2\left(x+2\right)\left(x+3\right)}=\frac{3\cdot3x\left(x-2\right)}{2\left(x+2\right)\left(x+3\right)\cdot3x\left(x-2\right)}=\frac{9x^2-18x}{6x\left(x-2\right)\left(x+2\right)\left(x+3\right)}\)

Bài 38:

Xét ΔABD và ΔACB có

\(\frac{AB}{AC}=\frac{AD}{AB}\left(\frac{10}{20}=\frac{5}{10}=\frac12\right)\)

góc BAD chung

Do đó: ΔABD~ΔACB

=>\(\hat{ABD}=\hat{ACB}\)

Bài 36:

Xét ΔABD và ΔBDC có

\(\frac{AB}{BD}=\frac{BD}{DC}\left(\frac48=\frac{8}{16}=\frac12\right)\)

\(\hat{ABD}=\hat{BDC}\) (hai góc so le trong, AB//CD)

Do đó: ΔABD~ΔBDC

=>\(\hat{BAD}=\hat{DBC}\)

ΔABD~ΔBDC

=>\(\frac{AD}{BC}=\frac{AB}{BD}=\frac48=\frac12\)

=>BC=2AD

35:

Xét ΔAMN và ΔACB có

\(\frac{AM}{AC}=\frac{AN}{AB}\left(\frac{10}{15}=\frac{8}{12}=\frac23\right)\)

góc MAN chung

Do đó: ΔAMN~ΔACB

=>\(\frac{MN}{CB}=\frac{AM}{AC}=\frac23\)

=>\(MN=18\cdot\frac23=12\left(\operatorname{cm}\right)\)

Bạn cần hỗ trợ bài nào nhỉ?

câu 3:

b) sửa đề: Tìm đa thức bậc ba P(x), bt rằng khi chia P(x) cho (x-1), cho (x-2) và (x-3) dư 6 và P(-1)=18

=> \(P\left(x\right)-6\) ⋮ \(\left(x-1\right)\left(x-2\right)\left(x-3\right)\)

đặt \(P\left(x\right)-6=a\left(x-1\right)\left(x-2\right)\left(x-3\right)\)

thay P(-1)=-18

=> \(P\left(-1\right)-6=a\left(-1-1\right)\left(-1-2\right)\left(-1-3\right)\)

\(-18-6=-24a\)

\(-24=-24a\)

=> \(a=1\)

vậy \(P\left(x\right)=1\left(x-1\right)\left(x-2\right)\left(x-3\right)+6\)

\(P\left(x\right)=\left(x^3-6x^2+11x-6\right)+6\)

\(P\left(x\right)=x^3-6x^2+11x\)

c) ta có: \(a_{k}=\frac{\left(2k+1\right)}{\left(k^2+k\right)^2}=\frac{\left(k+1\right)^2}{k^2\left(k+1\right)^2}-\frac{k^2}{k^2\left(k+1\right)^2}=\frac{1}{k^2}-\frac{1}{\left(k+1\right)^2}\)

=> \(S_{2018}=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+.\ldots+\frac{1}{2028^2}-\frac{1}{2029^2}\)

=> \(S_{2018}=1-\frac{1}{2019^2}\)

d) => \(\frac{\left(a+b-c\right)}{c}+2=\frac{\left(a+c-b\right)}{b}+2=\frac{\left(b+c-a\right)}{a}+2\)

\(\Rightarrow\frac{\left(a+b+c\right)}{c}=\frac{\left(a+b+c\right)}{b}=\frac{\left(a+b+c\right)}{a}\)

TH1: \(a+b+c=0\)

=> \(a+b=-c\)

\(b+c=-a\)

\(c+a=-b\)

=> \(P=\frac{\left(a+b\right)}{a}\cdot\frac{\left(b+c\right)}{b}\cdot\frac{\left(c+a\right)}{c}=-\frac{abc}{abc}=-1\)

TH2: \(a+b+c\) ≠0

=> \(\frac{\left(a+b+c\right)}{c}=\frac{\left(a+b+c\right)}{b}=\frac{\left(a+b+c\right)}{a}=\frac{3\left(a+b+c\right)}{\left(a+b+c\right)}=3\)

=> \(a+b+c=3c\)

\(a+b=2c\)

CMTT: \(b+c=2a\)

\(a+c=2b\)

thay vào P ta có:

\(P=\frac{\left(a+b\right)}{a}\cdot\frac{\left(b+c\right)}{b}\cdot\frac{\left(a+c\right)}{c}=\frac{2c}{a}\cdot\frac{2a}{b}\cdot\frac{2b}{c}=\frac{8abc}{abc}=8\)

câu 4:

a) vì AN//FM và AM//NF

=> ANFM là hình bình hành

xét tam giác ABM và tam giác ADM có:

góc ADM= góc ABM= 90 độ

AD=AB

BM=ND

=> △ABM=△AND(c.g.c)

=> AN=AM

=> AMFN là hình thoi

ta có góc MAN= góc MAD + góc MAD

mà góc MAD= góc BAM

=> góc MAN= góc BAM + góc MAD= 90 độ

=> AMFN là hình vuông

b) kẻ FH⊥BC tại H và FK⊥CD tại K

=> CHFK là hình chữ nhật

ta có góc HFM+góc MFK= 90 độ

mà góc NFK+ góc MFK= 90 độ

=> góc MFH= góc NFK

xét tam giác FNK và tam giác FMH có:

góc MFH= góc NFK

góc FHM= góc FKN= 90 độ

FN=FM

=> △FNK=△FMH(cg-gn)

=> FH=FK

=> CHFK là hình vuông

=> CF là phân giác góc HCK

=> F thuộc góc MCN

vì ABCD là hình vuông

=> góc ACB= 45 độ

vì CHFK là hình vuông

=> góc FCH= 45 độ

=> góc ACF= 180 độ- 45 độ- 45 độ= 90 độ

c) ta có ANFM là hình vuông

=> O là giao của AF và NM

=> O là trung điểm NM

=> \(OA=\frac12MN\)

mà xét tam giác CMN có CO là trung tuyến

=> \(OC=\frac12MN\)

=> \(OA=OC\)

=> C ∈ đường trung bình của của AC

mà DB vừa ⊥ AC và cắt trung điểm của nó tại AC

=> DB là đường trung bình của AC

=> O,D,B thẳng hàng

ta có BD⊥AC

mà FC⊥AC( do góc ACF= 90 độ)

=> BD//CF

=> tứ giác BOFC là hình thang

câu 5:

ta có trong tam giác AMB

=> AM+MB>AB=a

CMTT: => MC+MA > a

MB+MC>a

=> \(2\left(MA+MC+MB\right)>3a\)

=> \(MA+MB+MC>\frac{3a}{2}\)

mà 3> \(\sqrt3\)

=> \(MA+MB+MC>\frac{a\sqrt3}{2}\) (đpcm)

1B:

a: \(x^2+2xy+x+2y\)

=x(x+2y)+(x+2y)

=(x+2y)(x+1)

b: \(2xy+yz+2x+z\)

=y(2x+z)+(2x+z)

=(2x+z)(y+1)

c: \(y^2-2y-z^2-2z\)

\(=\left(y^2-z^2\right)-2\left(y+z\right)\)

=(y+z)(y-z)-2(y+z)

=(y+z)(y-z-2)

d: \(x^3-x-y+y^3\)

\(=\left(x^3+y^3\right)-\left(x+y\right)\)

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

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

2A:

a: \(x^2-2x+1-y^2\)

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

=(x-1-y)(x-1+y)

b: \(x^2-y^2+4y-4\)

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

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

=(x-y+2)(x+y-2)

c: \(y^2+6y-4z^2+9\)

\(=\left(y^2+6y+9\right)-\left(2z\right)^2\)

\(=\left(y+3\right)^2-\left(2z\right)^2=\left(y+3+2z\right)\left(y+3-2z\right)\)

d: \(x^2-y^2+10yz-25z^2\)

\(=x^2-\left(y^2-10yz+25z^2\right)\)

\(=x^2-\left(y-5z\right)^2=\left(x-y+5z\right)\left(x+y-5z\right)\)

2B:

a: \(4x^2-4x+1-25y^2\)

\(=\left(4x^2-4x+1\right)-\left(5y\right)^2\)

\(=\left(2x-1\right)^2-\left(5y\right)^2=\left(2x-1-5y\right)\left(2x-1+5y\right)\)

b: \(9y^2-z^2+6z-9\)

\(=\left(3y\right)^2-\left(z^2-6z+9\right)\)

\(=\left(3y\right)^2-\left(z-3\right)^2\)

=(3y-z+3)(3y+z-3)

c: \(x^2-4z^2+4x+4\)

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

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

=(x+2+2z)(x+2-2z)

d: \(4x^2-y^2+4xz+z^2\)

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

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

=(2x+z-y)(2x+z+y)

3A:

a: \(x^2-2xy+y^2-a^2+2ab-b^2\)

\(=\left(x^2-2xy+y^2\right)-\left(a^2-2ab+b^2\right)\)

\(=\left(x-y\right)^2-\left(a-b\right)^2\)

=(x-y-a+b)(x-y+a-b)

c: \(x^3+y^3+3x^2-3xy+3y^2\)

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

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