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Bài 1 : Với : \(x>0;x\ne1\)
\(P=\left(1+\frac{1}{\sqrt{x}-1}\right)\frac{1}{x-\sqrt{x}}=\left(\frac{\sqrt{x}}{\sqrt{x}-1}\right).\sqrt{x}\left(\sqrt{x}-1\right)=x\)
Thay vào ta được : \(P=x=25\)
Bài 2 :
a, Với \(x\ge0;x\ne1\)
\(A=\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{2}{\sqrt{x}+1}-\frac{2}{x-1}=\frac{x+\sqrt{x}-2\sqrt{x}+2-2}{x-1}\)
\(=\frac{x-\sqrt{x}}{x-1}=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}}{\sqrt{x}+1}\)
Thay x = 9 vào A ta được : \(\frac{3}{3+1}=\frac{3}{4}\)
\(ĐKXĐ:x\ne\pm1\)
a) \(A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}+\frac{4x^2}{1-x^2}\right):\frac{2x^2-2}{x^2-2x+1}\)
\(\Leftrightarrow A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}-\frac{4x^2}{x^2-1}\right):\frac{2\left(x^2-1\right)}{\left(x-1\right)^2}\)
\(\Leftrightarrow A=\frac{\left(x+1\right)^2-\left(x-1\right)^2-4x^2}{x^2-1}.\frac{\left(x-1\right)^2}{2\left(x^2-1\right)}\)
\(\Leftrightarrow A=\frac{x^2+2x+1-x^2+2x-1}{x^2-1}.\frac{\left(x-1\right)^2}{2\left(x^2-1\right)}\)
\(\Leftrightarrow A=\frac{4x-4x^2}{x^2-1}.\frac{\left(x-1\right)^2}{2\left(x^2-1\right)}\)
\(\Leftrightarrow A=\frac{-4x\left(x-1\right)^3}{2\left(x-1\right)^2\left(x+1\right)^2}\)
\(\Leftrightarrow A=\frac{-2x\left(x-1\right)}{\left(x+1\right)^2}\)
b) Thay x = -3 vào A, ta được :
\(A=\frac{\left(-2\right)\left(-3\right)\left(-3-1\right)}{\left(-3+1\right)^2}\)
\(\Leftrightarrow A=\frac{6.\left(-4\right)}{2^2}\)
\(\Leftrightarrow A=-6\)
c) Để A > -1
\(\Leftrightarrow-2x\left(x-1\right)>-\left(x+1\right)^2\)
\(\Leftrightarrow2x\left(x-1\right)< \left(x+1\right)^2\)
\(\Leftrightarrow2x^2-2x< x^2+2x+1\)
\(\Leftrightarrow x^2-4x-1< 0\)
\(\Leftrightarrow\left(x-2\right)^2-5< 0\)
\(\Leftrightarrow\left(x-2\right)^2< 5\)
Đoạn này bạn tự tìm giá trị x thỏa mãn là xong (Chú ý ĐKXĐ)
a) ĐKXĐ: \(x\ne-2;x\ne2\), rút gọn:
\(A=\left[\frac{3\left(x-2\right)-2x\left(x+2\right)+2\left(2x^2+3\right)}{2\left(x-2\right)\left(x+2\right)}\right]\div\frac{2x-1}{4\left(x-2\right)}\)
\(A=\frac{3x-6-2x^2-4x+4x^2+6}{2\left(x-2\right)\left(x+2\right)}\cdot\frac{4\left(x-2\right)}{2x-1}=\frac{4\left(2x^2-x\right)}{x\left(x+2\right)\left(2x-1\right)}=\frac{4x\left(2x-1\right)}{x\left(x+2\right)\left(2x-1\right)}=\frac{4}{x+2}\)
b) Ta có: \(\left|x-1\right|=3\Leftrightarrow\hept{\begin{cases}x-1=3\\x-1=-3\end{cases}\Leftrightarrow\hept{\begin{cases}x=4\left(n\right)\\x=-2\left(l\right)\end{cases}}}\)
=> Khi \(x=4\)thì \(A=\frac{4}{4+2}=\frac{4}{6}=\frac{2}{3}\)
c) \(A< 2\Leftrightarrow\frac{4}{x+2}< 2\Leftrightarrow4< 2x+4\Leftrightarrow0< 2x\Leftrightarrow x>0\)Vậy \(A< 2,\forall x>0\)
d) \(\left|A\right|=1\Leftrightarrow\left|\frac{4}{x+2}\right|=1\Leftrightarrow\hept{\begin{cases}\frac{4}{x+2}=1\\\frac{4}{x+2}=-1\end{cases}\Leftrightarrow\hept{\begin{cases}x=2\left(l\right)\\x=-6\left(n\right)\end{cases}}}\)Vậy \(\left|A\right|=1\)khi và chỉ khi x = -6
a) Ta có :A = \(\left(\frac{\left(x-1\right)^2}{3x+\left(x-1\right)^2}-\frac{1-2x^2+4x}{x^3-1}+\frac{1}{x-1}\right):\frac{x^2+x}{x^3+x}\)
ĐK: \(\hept{\begin{cases}x\ne0\\x\ne1\end{cases}}\)
A = \(\left(\frac{\left(x-1\right)^2}{x^2+x+1}-\frac{1-2x^2+4x}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{1}{x-1}\right):\frac{x\left(x+1\right)}{x\left(x^2+1\right)}\)
= \(\frac{\left(x-1\right)^3-1+2x^2-4x+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\)
= \(\frac{x^3-3x^2+3x-1+3x^2-3x}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\)
= \(\frac{x^3-1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}=1.\frac{x^2+1}{x+1}=\frac{x^2+1}{x+1}\)
b) Để A > - 1 <=> \(\frac{x^2+1}{x+1}>-1\)
<=> \(\frac{x^2+1}{x+1}+1>0\)
<=> \(\frac{x^2+x+2}{x+1}>0\)
Vì x2 + x + 2 >0 \(\forall x\)
=> A > 0 <=> x + 1 > 0 <=> x > -1
ĐKXĐ: \(x\ne-1\)
a)\(B=\frac{x^2-5x}{x^3+1}+\frac{x+2}{x^2-x+1}+\frac{1}{x+1}\)
\(=\frac{x^2-5x}{x^3+1}+\frac{\left(x+2\right)\left(x+1\right)}{\left(x^2-x+1\right)\left(x+1\right)}+\frac{x^2-x+1}{\left(x+1\right)\left(x^2-x+1\right)}\)
\(=\frac{x^2-5x}{x^3+1}+\frac{x^2+3x+2}{x^3+1}+\frac{x^2-x+1}{x^3+1}\)\(=\frac{3x^2-3x+3}{x^3+1}=\frac{3\left(x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\)
\(=\frac{3}{x+1}\)
b)B là số tự nhiên khi \(\frac{3}{x+1}\) là số tự nhiên<=>x+1 là ước dương của 3<=>x+1\(\in\left\{1;3\right\}\)<=>\(x\in\left\{0;2\right\}\)
ĐKXĐ \(x\ne0;x\ne1;x\ne-1\)
\(A=\frac{\left(x+1+1-x\right)}{\left(1-x^2\right)-\frac{5-x}{1-x^2}}:\frac{\left(1-2x\right)}{x^2-1}\)
\(A=\frac{\left(x-3\right)}{\left(1-x^2\right)}:\frac{\left(1-2x\right)}{\left(x^2-1\right)}\)
\(A=\frac{\left(3-x\right)}{\left(x^2-1\right)}:\frac{\left(1-2x\right)}{\left(x^2-1\right)}\)
\(A=\frac{\left(3x-2\right)}{1-2x}\)
\(a,ĐKXĐ:x\ne\pm1;x\ne\frac{1}{2}\)
\(A=\left(\frac{1}{x-1}+\frac{2}{x+1}-\frac{5-x}{1-x^{^2}}\right):\frac{1-2x}{x^2-1}\)
\(=\left(\frac{1}{x-1}+\frac{2}{x+1}+\frac{5-x}{\left(x-1\right)\left(x+1\right)}\right):\frac{1-2x}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x+1+2\left(x-1\right)+5-x}{\left(x-1\right)\left(x+1\right)}:\frac{1-2x}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{2x+4}{\left(x-1\right)\left(x+1\right)}.\frac{\left(x-1\right)\left(x+1\right)}{1-2x}\)
\(=\frac{2x+4}{1-2x}\)
\(b,Vớix\ne\pm1;x\ne\frac{1}{2}\)ta có \(A=\frac{2x+4}{1-2x}=\frac{-1\left(1-2x\right)+5}{1-2x}=-1+\frac{5}{1-2x}\)
Với x thuộc Z để A nguyên thì \(5⋮1-2x\Rightarrow1-2x\inƯ\left\{5\right\}=\left\{\pm1;\pm5\right\}\)
Với 1-2x=1 => x= 0(TMĐKXĐ)
với 1-2x=-1 => x=1(loại)
với 1-2x=5 => x=-2(tmđkxđ)
với 1-2x=-5 => x=3(tmđkxđ)
Vậy với \(x\in\left\{0;-2;-3\right\}\)thì A nguyên
a) A = \(\frac{3x^2+3x-3}{x^2+x-2}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\left(\frac{1}{1-x}-1\right)\)
A = \(\frac{3x^2+3x-3}{x^2+2x-x-2}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\left(\frac{1-1+x}{1-x}\right)\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\frac{x}{1-x}\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{x+1}{x+2}-\frac{x-2}{x-1}\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x-2\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{3x^2+3x-3-x^2+1-x^2+4}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x^2+3x+2}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x^2+2x+x+2}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x+1}{x-1}\) (Đk: \(x-1\ge0\) => x \(\ge\)1)
b) Ta có: A = \(\frac{x+1}{x-1}=\frac{\left(x-1\right)+2}{x-1}=1+\frac{2}{x-1}\)
Để A \(\in\)Z <=> 2 \(⋮\)x - 1
<=> x - 1 \(\in\)Ư(2) = {1; -1; 2; -2}
<=> x \(\in\){2; 0; 3; -1}
c) Ta có: A < 0
=> \(\frac{x+1}{x-1}< 0\)
=> \(\hept{\begin{cases}x+1< 0\\x-1>0\end{cases}}\) hoặc \(\hept{\begin{cases}x+1>0\\x-1< 0\end{cases}}\)
=> \(\hept{\begin{cases}x< -1\\x>1\end{cases}}\)(loại) hoặc \(\hept{\begin{cases}x>-1\\x< 1\end{cases}}\)
=> -1 < x < 1
Edogawa Conan
Thiếu dòng đầu \(ĐKXĐ:\hept{\begin{cases}x\ne1\\x\ne-2\\x\ne0\end{cases}}\)