câu c (x+3)(x^2-3x+9)-(54+x^3)=x^3+27-54-x^3

=27

Bài 3:

a: \(\frac{x}{x-3}+\frac{9-6x}{x^2-3x}\)

\(=\frac{x}{x-3}+\frac{-6x+9}{x\left(x-3\right)}\)

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

b: \(\frac{6x-3}{x}:\frac{4x^2-1}{3x^2}\)

\(=\frac{3\left(2x-1\right)}{x}\cdot\frac{3x^2}{\left(2x-1\right)\left(2x+1\right)}=\frac{3\cdot3x}{2x+1}=\frac{9x}{2x+1}\)

Bài 2:

a: \(\frac{x^3-x}{3x+3}\)

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

b: \(\frac{x^2+3xy}{x^2-9y^2}=\frac{x\left(x+3y\right)}{\left(x-3y\right)\left(x+3y\right)}=\frac{x}{x-3y}\)

Bài 1:

a: \(\frac{x^2-9}{2x+6}:\frac{3-x}{2}\)

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

b: \(\frac{2x}{x-y}-\frac{2y}{x-y}=\frac{2x-2y}{x-y}=\frac{2\left(x-y\right)}{x-y}=2\)

c: \(\frac{x+15}{x^2-9}+\frac{2}{x+3}\)

\(=\frac{x+15}{\left(x-3\right)\left(x+3\right)}+\frac{2}{x+3}\)

\(=\frac{x+15+2\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}=\frac{x+15+2x-6}{\left(x-3\right)\left(x+3\right)}=\frac{3x+9}{\left(x-3\right)\left(x+3\right)}\)

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

d: \(\frac{x+y}{2x-2y}-\frac{x-y}{2x+2y}-\frac{y^2+x^2}{y^2-x^2}\)

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

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

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

Hàm số xác định trên R khi và chỉ khi:

a.

\(\left(2m-4\right)x+m^2-9=0\) vô nghiệm

\(\Leftrightarrow\left\{{}\begin{matrix}2m-4=0\\m^2-9\ne0\end{matrix}\right.\) \(\Rightarrow m=2\)

b.

\(x^2-2\left(m-3\right)x+9=0\) vô nghiệm

\(\Leftrightarrow\Delta'=\left(m-3\right)^2-9< 0\)

\(\Leftrightarrow m^2-6m< 0\Rightarrow0< m< 6\)

c.

\(x^2+6x+2m-3>0\) với mọi x

\(\Leftrightarrow\Delta'=9-\left(2m-3\right)< 0\)

\(\Leftrightarrow m>6\)

e.

\(-x^2+6x+2m-3>0\) với mọi x

Mà \(a=-1< 0\Rightarrow\) không tồn tại m thỏa mãn

f.

\(x^2+2\left(m-1\right)x+2m-2>0\) với mọi x

\(\Leftrightarrow\Delta'=\left(m-1\right)^2-\left(2m-2\right)=m^2-4m+3< 0\)

\(\Leftrightarrow1< m< 3\)

a: Ta có: \(\left(2x+3\right)^2+\left(2x-3\right)^2-2\left(4x^2-9\right)\)

\(=4x^2+12x+9+4x^2-12x+9-8x^2+18\)

\(=36\)

Bài 2: 

a: \(\left(y^2+6x^2\right)\left(y^2-6x^2\right)=y^4-36x^4\)

b: \(\left(4x+5\right)\left(16x^2-20x+25\right)=\left(16x^2-25\right)\left(4x-5\right)\)

\(=64x^3-16x^2-100x+125\)

1: Đặt \(a=9x^2-6x\)

=>\(45x^2-30x=5\left(9x^2-6x\right)=5a\)

\(\sqrt{9x^2-6x+2}+\sqrt{45x^2-30x+9}=\sqrt{6x-9x^2+8}\)

=>\(\sqrt{a+2}+\sqrt{5a+9}=\sqrt{-a+8}\)

=>\(\sqrt{a+2}-1+\sqrt{5a+9}-2=\sqrt{-a+8}-3\)

=>\(\frac{a+2-1}{\sqrt{a+2}+1}+\frac{5a+9-4}{\sqrt{5a+9}+2}=\frac{-a+8-9}{\sqrt{-a+8}+3}\)

=>\(\left(a+1\right)\left(\frac{1}{\sqrt{a+2}+1}+\frac{5}{\sqrt{5a+9}+2}+\frac{1}{\sqrt{a+8}+3}\right)=0\)

=>a+1=0

=>a=-1

=>\(9x^2-6x=-1\)

=>\(9x^2-6x+1=0\)

=>\(\left(3x-1\right)^2=0\)

=>3x-1=0

=>3x=1

=>x=1/3

2: Đặt \(x^2-2x=a\)

\(\sqrt{2x^2-4x+3}+\sqrt{3x^2-6x+7}=2-x^2+2x\)

=>\(\sqrt{2\left(x^2-2x\right)+3}+\sqrt{3\left(x^2-2x\right)+7}=-\left(x^2-2x\right)+2\)

=>\(\sqrt{2a+3}+\sqrt{3a+7}=-a+2\)

=>\(\sqrt{2a+3}-1+\sqrt{3a+7}-2=-a+2-3\)

=>\(\frac{2a+2}{\sqrt{2a+3}+1}+\frac{3a+7-4}{\sqrt{3a+7}+2}=-a-1\)

=>\(\left(a+1\right)\left(\frac{2}{\sqrt{2a+3}+1}+\frac{3}{\sqrt{3a+7}+2}+1\right)=0\)

=>a+1=0

=>\(x^2-2x+1=0\)

=>\(\left(x-1\right)^2=0\)

=>x-1=0

=>x=1

a) 3x³ + 6x²y

= 3x².(x + 2y)

b) 2x³ - 6x²

= 2x².(x - 2)

c) 18x² - 20xy

= 2x.(9x - 10y)

d) xy + y² - x - y

= (xy + y²) - (x + y)

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

= (x + y)(y - 1)

e) (x²y² - 8)² - 1

= (x²y² - 8 - 1)(x²y² - 8 + 1)

= (x²y² - 9)(x²y² - 7)

= (xy - 3)(xy + 3)(x²y² - 7)

f) x² - 7x - 8

= x² - 8x + x - 8

= (x² - 8x) + (x - 8)

= x(x - 8) + (x - 8)

= (x - 8)(x + 1)

a: \(3x^3+6x^2y\)

\(=3x^2\cdot x+3x^2\cdot2y=3x^2\left(x+2y\right)\)

b: \(2x^3-6x^2=2x^2\cdot x-2x^2\cdot3=2x^2\left(x-3\right)\)

c: \(18x^2-20xy=2x\cdot9x-2x\cdot10y=2x\left(9x-10y\right)\)

d: \(xy+y^2-x-y\)

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

\(=\left(x+y\right)\left(y-1\right)\)

e: \(\left(x^2y^2-8\right)^2-1\)

\(=\left(x^2y^2-8-1\right)\left(x^2y^2-8+1\right)\)

\(=\left(x^2y^2-7\right)\left(x^2y^2-9\right)\)

\(=\left(x^2y^2-7\right)\left(xy-3\right)\left(xy+3\right)\)

f: \(x^2-7x-8\)

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

\(=x\left(x-8\right)+\left(x-8\right)=\left(x-8\right)\left(x+1\right)\)

g: \(10x^2\left(2x-y\right)+6xy\left(y-2x\right)\)

\(=2x\cdot\left(2x-y\right)\cdot5x-2x\cdot\left(2x-y\right)\cdot3y\)

\(=2x\left(2x-y\right)\left(5x-3y\right)\)

h: \(x^2-2x+1-y^2\)

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

\(=\left(x-1-y\right)\left(x-1+y\right)\)

i: \(2x\left(x+2\right)+x^2\left(-x-2\right)\)

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

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

\(=-\left(x^2-2\cdot x\cdot3+3^2\right)=-\left(x-3\right)^2\)

l: \(-2x^2+8xy-8y^2\)

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

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

m: \(3x^2+5x-3y^2-5y\)

\(=3\left(x^2-y^2\right)+5\left(x-y\right)\)

\(=3\left(x-y\right)\left(x+y\right)+5\left(x-y\right)\)

\(=\left(x-y\right)\left(3x+3y+5\right)\)

\(\frac{x^2-y^2+6x+9}{x+y+3}\)

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

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

\(=\frac{\left(x+3+y\right)\left(x+3-y\right)}{x+y+3}\)

\(=x+3-y\)

a: \(15xy^2z^3:3xyz^2=5yz\)

b: \(12x^4y^4:\left(-4x^4y^2\right)=-3y^2\)

c: \(\dfrac{-15x^2y^3z^2}{-6xz^2}=\dfrac{5}{2}xy^3\)

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

mình có 6ý mà