Bài 1:

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

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

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

b: Để A>0 thì 1-2x>0

=>2x<1

=>x<1/2

Chọn C

\(\lim\limits_{x\rightarrow0}\dfrac{2\left(\sqrt{3x+1}-1\right)}{x}=\lim\limits_{x\rightarrow0}\dfrac{6x}{x\left(\sqrt{3x+1}+1\right)}=\lim\limits_{x\rightarrow0}\dfrac{6}{\sqrt{3x+1}+1}=3\)

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

\(\Rightarrow I-J=6\)

a)PT: \(x^2-2\left(m-1\right)x+2m-5=0\)

\(\Rightarrow\Delta=\left(-2\left(m-1\right)\right)^2-4.1.\left(2m-5\right)\\ =4m^2-16m+24=\left(2m-4\right)^2+8\ge8\left(\forall m\in R\right)\)

Vậy phương trình luôn có hai nghiệm phân biệt \(x_1,x_2\) với mọi m.

p/s: phần (b) mình sẽ giúp bạn trả lời sau nha!

a) \(36x^2-49=0\)

\(\Leftrightarrow\left(6x\right)^2-7^2=0\)

\(\Leftrightarrow\left(6x-7\right)\left(6x+7\right)=0\)

\(TH_1:6x-7=0\) \(TH_2:6x+7=0\)

\(\Leftrightarrow6x=7\) \(\Leftrightarrow6x=-7\)

\(\Leftrightarrow x=\dfrac{7}{6}\) \(\Leftrightarrow x=-\dfrac{7}{6}\)

Vậy pt có tập nghiệm \(S=\left\{\dfrac{7}{6};-\dfrac{7}{6}\right\}\)

Bài 2

a) 36x²-49=0

⇔ (6x)²-49=0

⇔(6x-7).(6x+7)=0

TH1: 6x-7=0 TH2: 6x+7=0

⇔6x=7 ⇔6x=-7

⇔x=7/6 ⇔x=-7/6

1,

\(x^2+y^2+y^2=14\)

\(\Rightarrow\left(x+y+z\right)^2-2xy-2yz-2zx=14\)

\(\Rightarrow-2\left(xy+yz+zx\right)=14\)

\(\Rightarrow xy+yz+zx=-7\)

\(\Rightarrow\left(xy+yz+zx\right)^2=49\)

\(\Leftrightarrow x^2y^2+y^2z^2+z^2x^2+2x^2yz+2xy^2z+2xyz^2=49\)

\(\Leftrightarrow x^2y^2+y^2z^2+z^2x^2+2xyz\left(x+y+z\right)=49\)

\(\Leftrightarrow x^2y^2+y^2z^2+z^2x^2=49\)

Ta có: \(x^4+y^4+z^4\)

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

\(=14^2-2\left(x^2y^2+y^2z^2+z^2x^2\right)\)

\(=14^2-2.49\)

\(=196-98\)

\(=98\)

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

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

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

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

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

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

b: Để A<0 thì \(\sqrt{x}-1< 0\)

=>0<x<1

1 ) \(\left(x-4\right)^2-25=0\)

\(\Leftrightarrow\left(x-4-5\right)\left(x-4+5\right)=0\)

\(\Leftrightarrow\left(x-9\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=9\\x=-1\end{matrix}\right.\)

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

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

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

\(\Leftrightarrow x=2.\)

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

\(\Leftrightarrow\left(x^2-4\right)\left(2x+3-x+1\right)=0\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-2\\x=-4\end{matrix}\right.\)

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

\(\Leftrightarrow\left(x+1\right)\left(x-1-2+3x\right)=0\)

\(\Leftrightarrow\left(x+1\right)\left(4x-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=\dfrac{3}{4}\end{matrix}\right.\)

5 ) \(x^3+x^2+x+1=0\)

\(\Leftrightarrow\left(x^2+1\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2=-1\left(loại\right)\\x=-1.\end{matrix}\right.\)

6 ) \(x^3+x^2-x-1=0\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-1\end{matrix}\right.\)

7 ) \(2x^3+3x^2+6x+5=0\)

\(\Leftrightarrow2x^3+2x^2+x^2+x+5x+5=0\)

\(\Leftrightarrow2x^2\left(x+1\right)+x\left(x+1\right)+5\left(x+1\right)=0\)

\(\Leftrightarrow\left(2x^2+x+5\right)\left(x+1\right)=0\)

\(\Leftrightarrow x=-1.\)

8 ) \(x^4-4x^3-19x^2+106x-120=0\)

\(\Leftrightarrow x^4-4x^3-19x^2+76x+30x-120=0\)

\(\Leftrightarrow x^3\left(x-4\right)-19x\left(x-4\right)+30\left(x-4\right)=0\)

\(\Leftrightarrow\left(x^3-19x+30\right)\left(x-4\right)=0\)

\(\Leftrightarrow\left(x^3-8-19x+38\right)\left(x-4\right)\)

\(\Leftrightarrow\left(x-2\right)\left(x^2+4x+23\right)\left(x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=4\end{matrix}\right.\)

9 ) \(\left(x^2-3x+2\right)\left(x^2+15x+56\right)+8=0\)

\(\Leftrightarrow\left(x-2\right)\left(x-1\right)\left(x+7\right)\left(x+8\right)+8=0\)

\(\Leftrightarrow\left(x^2+7x-x-7\right)\left(x^2+8x-2x-16\right)+8=0\)

\(\Leftrightarrow\left(x^2+6x-7\right)\left(x^2+6x-16\right)+8=0\)

Đặt \(x^2+6x-7=t\)

\(\Leftrightarrow t\left(t-9\right)+8=0\)

\(\Leftrightarrow t^2-9t+8=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=8\\t=1\end{matrix}\right.\)

Khi t = 8 \(\Leftrightarrow x^2+6x-7=8\Leftrightarrow x^2+6x-15\Leftrightarrow\left[{}\begin{matrix}x=-3+2\sqrt{6}\\x=-3-2\sqrt{6}\end{matrix}\right.\)

Khi t = 1 \(\Leftrightarrow x^2+6x-7=1\Leftrightarrow x^2+6x-8=0\Leftrightarrow\left[{}\begin{matrix}x=-3+\sqrt{17}\\x=-3-\sqrt{17}\end{matrix}\right.\)

Vậy ........