ĐKXĐ: \(x\ge0;x\ne1\)

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

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

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

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

a; \(x\) + 6 ⋮ \(x\) + 1 (\(x\) ≠ - 1)

   \(x\) + 1 + 5 ⋮ \(x\) + 1

    \(x\) + 1 \(\in\) Ư(5) = {-5; -1; 1; 5}

    \(x\)       \(\in\) {-6; -2; 0; 4}

   \(x\) + 6 ⋮ \(x\) + (-1)     (\(x\) ≠ 1)

   \(x\) + - 1 + 7  ⋮ \(x\) - 1

                  7 ⋮ \(x\) - 1

 \(x\) - 1  \(\in\) Ư(7) = {-7; -1; 1; 7}

 \(x\)        \(\in\) {-6; 0; 2; 8}

b;   \(x\) + 6 ⋮ \(x\) - 2 (đk \(x\) ≠ 2)

 \(x\) - 2 + 8 ⋮ \(x\) - 2

            8 ⋮  \(x\) - 2

\(x\) - 2 \(\in\) Ư(8) = {-8; -4; -2; -1; 1; 2; 4; 8}

\(x\) \(\in\) {-6; -2; 0; 1; 3; 4; 10}

\(x\) + 6 ⋮ \(x\) + (-2)

\(x\) + 6  ⋮ \(x\) - 2

giống với ý trên

c; \(x\) + 7 ⋮ \(x\) - 2 (đk \(x\) ≠ 2)

    \(x\) - 2 + 9 ⋮ \(x\) - 2

                9 ⋮ \(x\) - 2

\(x\) - 2 \(\in\) {-9; -3; -1; 1; 3; 9}

\(x\)  \(\in\) {-7; -1; 1; 3; 5; 11}

       \(x\) + 7 \(⋮\) \(x\) + 2 (đk \(x\) ≠ -2}

  \(x\) + 2 + 5 \(⋮\) \(x\) + 2

              5 ⋮ \(x\) + 2

\(x\) + 2 \(\in\) Ư(5) = {-5; -1; 1; 5}

\(x\) \(\in\) {-7; -3; -1; 3}

Chọn C

4.

Gọi H là chân đường cao kẻ từ C xuống đường thẳng d.

Ta có: \(CH=d\left(C;d\right)=\dfrac{\left|-3.2-4.5+4\right|}{\sqrt{3^2+4^2}}=\dfrac{22}{5}\)

Khi đó: \(S_{ABC}=\dfrac{1}{2}CH.AB=\dfrac{1}{2}.\dfrac{22}{5}.AB=15\Rightarrow AB=\dfrac{75}{11}\)

\(\Rightarrow IA=IB=\dfrac{75}{22}\)

Gọi \(A=\left(4m;3m+1\right)\) là điểm cần tìm.

Ta có: \(IA=\dfrac{75}{22}\Leftrightarrow\sqrt{\left(4m-2\right)^2+\left(3m-\dfrac{3}{2}\right)^2}=\dfrac{75}{22}\)

\(\Leftrightarrow\sqrt{25m^2-25m+\dfrac{25}{4}}=\dfrac{75}{22}\)

\(\Leftrightarrow\left|m-\dfrac{1}{2}\right|=\dfrac{15}{22}\)

\(\Leftrightarrow\left[{}\begin{matrix}m-\dfrac{1}{2}=\dfrac{15}{22}\\m-\dfrac{1}{2}=-\dfrac{15}{22}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m=\dfrac{13}{11}\\m=-\dfrac{2}{11}\end{matrix}\right.\)

\(m=\dfrac{13}{11}\Rightarrow A=\left(\dfrac{52}{11};\dfrac{50}{11}\right)\Rightarrow B=\left(-\dfrac{8}{11};\dfrac{5}{11}\right)\)

Vậy \(A=\left(\dfrac{52}{11};\dfrac{50}{11}\right);B=\left(-\dfrac{8}{11};\dfrac{5}{11}\right)\)

1.

\(P=\left(m;m+1\right)\) là điểm cần tìm 

\(\Rightarrow NP=\sqrt{\left(m-3\right)^2+m^2}=\sqrt{2m^2-6m+9}\)

Ta có: \(NM=NP\)

\(\Leftrightarrow\sqrt{\left(-1-3\right)^2+\left(2-1\right)^2}=\sqrt{2m^2-6m+9}\)

\(\Leftrightarrow m^2-3m-4=0\)

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

\(\Rightarrow\left[{}\begin{matrix}P=\left(4;5\right)\\P=\left(-1;0\right)\end{matrix}\right.\)

Vậy \(P=\left(4;5\right)\) hoặc \(P=\left(-1;0\right)\)

2.

\(tanx=-2\Leftrightarrow\dfrac{sinx}{cosx}=-2\Leftrightarrow sinx=-2cosx\)

Khi đó:

\(A=\dfrac{sin^2x+3sinx.cosx-cos^2x+1}{3sin^2x+4sinx.cosx+5cos^2x-2}\)

\(=\dfrac{2sin^2x+3sinx.cosx}{2\left(sin^2x-1\right)+sin^2x+5cos^2x+4sinx.cosx}\)

\(=\dfrac{2sin^2x+3sinx.cosx}{sin^2x+3cos^2x+4sinx.cosx}\)

\(=\dfrac{8cos^2x-3.2cos^2x}{4cos^2x+3cos^2x-4.2cos^2x}\)

\(=\dfrac{2cos^2x}{-cos^2x}=-2\)

Bài 1

a) A = \(\dfrac{x}{x-4}+\dfrac{1}{\sqrt{x}+2}+\dfrac{1}{\sqrt{x}-2}\) (ĐK: x ≥ 0; x ≠ 4)

↔ A = \(\dfrac{x+2-\sqrt{x}+\sqrt{x}+2}{x-4}\)

↔ A = \(\dfrac{x+4}{x-4}\)

Để A = 2 ↔ \(\dfrac{x+4}{x-4}\) = 2 (ĐK: x ≠ 4)

→  \(x+4=2\left(x-4\right)\)

↔  \(2x-x=4+8\)

↔ \(x=12\)

Vậy x = 12 thì A = 2

b) Để A < 1

↔ \(\dfrac{x+4}{x-4}\) < 1

→  \(x+4\) < \(x-4\)

↔ 0x < -8 (vô lý)

Vậy không có giá trị của x nào thỏa mãn A < 1

a,\(\dfrac{3x+5}{x-2}=3+\dfrac{11}{x-2}\)

\((3x+5)\vdots (x-2)\) \(\Rightarrow\)\(\dfrac{3x+5}{x-2}\)nguyên \(\Rightarrow \dfrac{11}{x-2}\)nguyên

\(\Rightarrow 11\vdots(x-2)\Rightarrow (x-2)\in Ư(11)=\{\pm1;\pm11\}\)

\(\Rightarrow x\in\{-9;1;3;13\}\)

b,\(\dfrac{2-4x}{x-1}=-4-\dfrac{2}{x-1}\)

\((2-4x)\vdots(x-1)\Rightarrow \dfrac{2-4x}{x-1}\)nguyên\(\Rightarrow \dfrac{2}{x-1}\)nguyên

\(\Rightarrow 2\vdots(x-1)\Rightarrow (x-1)\inƯ(2)=\{\pm1;\pm2\}\\\Rightarrow x\in\{-1;0;2;3\}\)

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

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

\(\Rightarrow 2\vdots(x-1)\Rightarrow (x-1)\inƯ(2)=\{\pm1;\pm2\}\\\Rightarrow x\in\{-1;0;2;3\}\)

d,\(\dfrac{x^{2}+2x+4}{x+1}=\dfrac{(x+1)^{2}+3}{x+1}=x+1+\dfrac{3}{x+1}\)

\((x^{2}+2x+4)\vdots(x+1)\Rightarrow \dfrac{x^{2}+2x+4}{x+1}\in Z\Rightarrow \dfrac{3}{x+1}\in Z\\\Rightarrow3\vdots(x+1)\Rightarrow (x+1)\in Ư(3)=\{\pm1;\pm3\}\\\Rightarrow x\in\{-4;-2;0;2\}\)

Giúp mình với

p₌(n+2).(n²+n-5)