a) \(A=x^2-6x+10=\left(x^2-6x+9\right)+1=\left(x-3\right)^2+1\ge1\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow x=3\). \(min_A=1\)

b) \(B=3x^2+x-2=3\left(x^2+\dfrac{1}{3}x-\dfrac{2}{3}\right)=3\left(x^2+\dfrac{1}{3}x+\dfrac{1}{36}-\dfrac{25}{36}\right)=3\left(x+\dfrac{1}{6}\right)^2-\dfrac{25}{12}\ge\dfrac{-25}{12}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow x=-\dfrac{1}{6}\). \(min_B=\dfrac{-25}{12}\)

c) \(C=\dfrac{4}{x^2}-\dfrac{3}{x}-1=\left(\dfrac{4}{x^2}-\dfrac{3}{x}+\dfrac{9}{16}\right)-\dfrac{25}{16}=\left(\dfrac{2}{x}+\dfrac{2}{3}\right)^2-\dfrac{25}{16}\ge\dfrac{-25}{16}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow x=-3\). \(min_C=\dfrac{-25}{16}\)

d) \(D=x^2+y^2-x+3y+7=\left(x^2-x+\dfrac{1}{4}\right)+\left(y^2+3y+\dfrac{9}{4}\right)+\dfrac{9}{2}=\left(x-\dfrac{1}{2}\right)^2+\left(y+\dfrac{3}{2}\right)^2+\dfrac{9}{2}\ge\dfrac{9}{2}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=\dfrac{-3}{2}\end{matrix}\right.\). \(min_D=\dfrac{9}{2}\)

a: ĐKXĐ: \(x^2+y^2\ne0\)

=>\(\left[{}\begin{matrix}x^2\ne0\\y^2\ne0\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x\ne0\\y\ne0\end{matrix}\right.\)

b: ĐKXĐ: \(x^2-2x+1\ne0\)

=>\(\left(x-1\right)^2\ne0\)

=>\(x-1\ne0\)

=>\(x\ne1\)

c: ĐKXĐ: \(x^2+6x+10\ne0\)

=>\(x^2+6x+9+1\ne0\)

=>\(\left(x+3\right)^2+1\ne0\)(luôn đúng)

d:ĐKXĐ: \(\left(x+3\right)^2+\left(y-2\right)^2\ne0\)

=>\(\left[{}\begin{matrix}x+3\ne0\\y-2\ne0\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x\ne-3\\y\ne2\end{matrix}\right.\)

Ko cần biet vi ko biet ang ang

\(\dfrac{1}{2022}\) \(\times\) \(\dfrac{2}{5}\) + \(\dfrac{1}{2022}\) \(\times\) \(\dfrac{7}{5}\) - \(\dfrac{1}{2022}\) \(\times\) \(\dfrac{8}{10}\)

= \(\dfrac{1}{2022}\) \(\times\) ( \(\dfrac{2}{5}\) + \(\dfrac{7}{5}\) - \(\dfrac{8}{10}\))

= \(\dfrac{1}{2022}\) \(\times\) ( \(\dfrac{9}{5}\) - \(\dfrac{4}{5}\))

= \(\dfrac{1}{2022}\) \(\times\) \(\dfrac{5}{5}\)

=  \(\dfrac{1}{2022}\times1\)

= \(\dfrac{1}{2022}\)

1) ĐKXĐ: \(x^2+2x-3\ge0\Leftrightarrow\left(x+1\right)^2\ge4\)

\(\Leftrightarrow\left[{}\begin{matrix}x+1\ge2\\x+1\le-2\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x\ge1\\x\le-3\end{matrix}\right.\)

2) ĐKXĐ: \(2x^2+5x+3\ge0\Leftrightarrow2\left(x+\dfrac{5}{4}\right)^2\ge\dfrac{1}{8}\Leftrightarrow\left(x+\dfrac{5}{4}\right)^2\ge\dfrac{1}{16}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{5}{4}\ge\dfrac{1}{4}\\x+\dfrac{5}{4}\le-\dfrac{1}{4}\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x\ge-1\\x\le-\dfrac{3}{2}\end{matrix}\right.\)

3) ĐKXĐ: \(x-1>0\Leftrightarrow x>1\)

4) ĐKXĐ: \(x-3< 0\Leftrightarrow x< 3\)

5) ĐKXĐ: \(x+2< 0\Leftrightarrow x< -2\)

6) ĐKXĐ: \(2a-1>0\Leftrightarrow a>\dfrac{1}{2}\)

Với `x \ne -5,x \ne -1` có:

`A=[x+2]/[x+5]+[-5x-1]/[x^2+6x+5]-1/[1+x]`

`A=[(x+2)(x+1)-5x-1-(x+5)]/[(x+5)(x+1)]`

`A=[x^2+x+2x+2-5x-1-x-5]/[(x+5)(x+1)]`

`A=[x^2-3x-4]/[(x+5)(x+1)]`

`A=[(x-4)(x+1)]/[(x+5)(x+1)]`

`A=[x-4]/[x+5]`

\(=\dfrac{x+2}{x+5}+\dfrac{-5x-1}{x^2+x+5x+5}-\dfrac{1}{x+1}\\ =\dfrac{x+2}{x+5}+\dfrac{-5x-1}{\left(x^2+x\right)+\left(5x+5\right)}-\dfrac{1}{x+1}\\ =\dfrac{\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}+\dfrac{-5x-1}{x\left(x+1\right)+5\left(x+1\right)}-\dfrac{x+5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}+\dfrac{-5x-1}{\left(x+1\right)\left(x+5\right)}-\dfrac{x+5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2+2x+x+2-5x-1-x-5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2-3x-4}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2+x-4x-4}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x^2+x\right)-\left(4x+4\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x\left(x+1\right)-4\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x+1\right)\left(x-4\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x-4}{x+5}\)

`a)` Thay `x=2` vào `B` có: `B=[-10]/[2-4]=5`

`b)` Với `x ne -1;x ne -5` có:

`A=[(x+2)(x+1)-5x-1-(x+5)]/[(x+1)(x+5)]`

`A=[x^2+x+2x+2-5x-1-x-5]/[(x+1)(x+5)]`

`A=[x^2-3x-4]/[(x+1)(x+5)]`

`A=[(x+1)(x-4)]/[(x+1)(x+5)]`

`A=[x-4]/[x+5]`

`c)` Với `x ne -5; x ne -1; x ne 4` có:

`P=A.B=[x-4]/[x+5].[-10]/[x-4]`

           `=[-10]/[x+5]`

Để `P` nguyên `<=>[-10]/[x+5] in ZZ`

    `=>x+5 in Ư_{-10}`

Mà `Ư_{-10}={+-1;+-2;+-5;+-10}`

`=>x={-4;-6;-3;-7;0;-10;5;-15}` (t/m đk)

b) \(\dfrac{x}{x-3}\) + \(\dfrac{-9}{x^2-3x}\)

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

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

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

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

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

#Fiona

c) \(\dfrac{3}{x-3}-\dfrac{6x}{9-x^2}+\dfrac{x}{x+3}\)

=\(\dfrac{3}{x-3}\) - \(\dfrac{6x}{3^2-x^2}\) + \(\dfrac{x}{x+3}\)

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

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

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

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

=\(\dfrac{3^2+2.3x+x^2}{\left(x-3\right)\left(x+3\right)}\)

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

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

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

#Fiona 

Tick đúng giúp mình nhaa<3

d)\(\dfrac{5x+10}{4x-8}\).\(\dfrac{x-2}{x+2}\)

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

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

=\(\dfrac{5}{4}\)

#Fiona

Tick đúng giúp mikk nhaa

a: \(x-3\left(2x-6\right)=21-\left(5x+3\right)\)

=>\(x-6x+18=21-5x-3\)

=>18=18(luôn đúng)

=>\(x\in R\)

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

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

=>2x-5=2x+2

=>-7=0(vô lý)

=>\(x\in\varnothing\)

c: \(\dfrac{9x+4}{6}=1-\dfrac{3x-5}{9}\)

=>\(\dfrac{3\left(9x+4\right)}{18}=\dfrac{18}{18}-\dfrac{2\left(3x-5\right)}{18}\)

=>3(9x+4)=18-2(3x-5)

=>27x+12=18-6x+10

=>27x+12=-6x+28

=>33x=16

=>\(x=\dfrac{16}{33}\left(nhận\right)\)

d: ĐKXĐ: \(x\notin\left\{2;5\right\}\)

\(\dfrac{6x+1}{x^2-7x+10}+\dfrac{5}{x-2}=\dfrac{3}{x-5}\)

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

=>\(6x+1+5\left(x-5\right)=3\left(x-2\right)\)

=>6x+1+5x-25=3x-6

=>11x-24=3x-6

=>8x=18

=>\(x=\dfrac{9}{4}\left(nhận\right)\)

a: 

=>

=>18=18(luôn đúng)

=>

b: 

=>

=>2x-5=2x+2

=>-7=0(vô lý)

=>x∈∅

c: 3(9x+4)18=1818−2(3x−5)18

=>3(9x+4)=18-2(3x-5)

=>27x+12=18-6x+10

=>27x+12=-6x+28

=>33x=16

=>

=>

f: \(=\dfrac{5x-3-x+3}{4x^2y}=\dfrac{4x}{4x^2y}=\dfrac{1}{xy}\)

g: \(=\dfrac{3x+10-x-4}{x+3}=\dfrac{2x+6}{x+3}=2\)

h: \(=\dfrac{4-2+x}{x-1}=\dfrac{x+2}{x-1}\)

n: \(=\dfrac{3x-x+6}{x\left(x+3\right)}=\dfrac{2\left(x+3\right)}{x\left(x+3\right)}=\dfrac{2}{x}\)

p: \(=\dfrac{x^2-9-x^2+9}{x\left(x-3\right)}=0\)

k: \(=\dfrac{x-2x-4+x-2}{\left(x+2\right)\left(x-2\right)}=\dfrac{-6}{x^2-4}\)

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