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a) \(ĐKXĐ:x\ne-3;x\ne2\)
b) \(P=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x-2\right)\left(x+3\right)}-\frac{x+3}{\left(x-2\right)\left(x+3\right)}\)
\(P=\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}\)
\(P=\frac{x^2-x-12}{\left(x+3\right)\left(x-2\right)}\)
\(P=\frac{\left(x+3\right)\left(x-4\right)}{\left(x+3\right)\left(x-2\right)}\)
vậy \(P=\frac{x-4}{x-2}\)
\(P=\frac{-3}{4}\) \(\Leftrightarrow\frac{x-4}{x-2}=\frac{-3}{4}\)
\(\Leftrightarrow4\left(x-4\right)=-3.\left(x-2\right)\)
\(\Leftrightarrow4x-16=-3x+6\)
\(\Leftrightarrow7x=22\)
\(\Leftrightarrow x=\frac{22}{7}\)
c) \(P\in Z\Leftrightarrow\frac{x-4}{x-2}\in Z\)
\(\frac{x-2-6}{x-2}=1-\frac{6}{x-2}\in Z\)
mà \(1\in Z\Rightarrow\left(x-2\right)\inƯ\left(6\right)\in\left(\pm1;\pm2;\pm3;\pm6\right)\)
mà theo ĐKXĐ: \(\Rightarrow\in\left(\pm1;-2;3;\pm6\right)\)
thay mấy cái kia vào rồi tìm \(x\)
d) \(x^2-9=0\Rightarrow x^2=9\Rightarrow x=\pm3\)
khi \(x=3\Rightarrow P=\frac{3-4}{3-2}=-1\)
khi \(x=-3\Rightarrow P=\frac{-3-4}{-3-2}=\frac{-7}{-5}=\frac{7}{5}\)
Dài quá trôi hết đề khỏi màn hình: nhìn thấy câu nào giải cấu ấy
Bài 4:
\(A=\frac{\left(x-1\right)+\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}-\frac{2}{\left(x+1\right)\left(x-1\right)}=\frac{2\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}\)
a) DK x khác +-1
b) \(dk\left(a\right)\Rightarrow A=\frac{2}{\left(x+1\right)}\)
c) x+1 phải thuộc Ước của 2=> x=(-3,-2,0))
1. a) Biểu thức a có nghĩa \(\Leftrightarrow\hept{\begin{cases}x+2\ne0\\x^2-4\ne0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x+2\ne0\\x-2\ne0\\x+2\ne0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x\ne-2\\x\ne2\end{cases}}\)
Vậy vs \(x\ne2,x\ne-2\) thì bt a có nghĩa
b) \(A=\frac{x}{x+2}+\frac{4-2x}{\left(x-2\right)\left(x+2\right)}\)
\(=\frac{x\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}+\frac{4-2x}{\left(x+2\right)\left(x-2\right)}\)
\(=\frac{x^2-2x+4-2x}{\left(x+2\right)\left(x-2\right)}\)
\(=\frac{x^2-4x+4}{\left(x+2\right)\left(x-2\right)}\)
\(=\frac{\left(x-2\right)^2}{\left(x+2\right)\left(x-2\right)}\)
\(=\frac{x-2}{x+2}\)
c) \(A=0\Leftrightarrow\frac{x-2}{x+2}=0\)
\(\Leftrightarrow x-2=\left(x+2\right).0\)
\(\Leftrightarrow x-2=0\)
\(\Leftrightarrow x=2\)(ko thỏa mãn điều kiện )
=> ko có gía trị nào của x để A=0
Bài làm
\(P=\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\)
a) ĐKXĐ : \(\hept{\begin{cases}x\ne-3\\x\ne2\end{cases}}\)
\(=\frac{x+2}{x+3}-\frac{5}{x^2+3x-2x-6}-\frac{1}{x-2}\)
\(=\frac{x+2}{x+3}-\frac{5}{x\left(x+3\right)-2\left(x+3\right)}-\frac{1}{x-2}\)
\(=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{x+3}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-4}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{x+3}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-x-12}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-4x+3x-12}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x\left(x-4\right)+3\left(x-4\right)}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{\left(x-4\right)\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}=\frac{x-4}{x-2}\)
b) x2 - 9 = 0 <=> ( x - 3 )( x + 3 ) = 0
<=> \(\orbr{\begin{cases}x=3\left(nhan\right)\\x=-3\left(loai\right)\end{cases}}\)
x = 3 => \(P=\frac{3-4}{3-2}=-1\)
c) \(P=\frac{x-4}{x-2}=\frac{x-2-2}{x-2}=1-\frac{2}{x-2}\)
Để P đạt giá trị nguyên => \(\frac{2}{x-2}\)nguyên
=> \(2⋮x-2\)
=> \(x-2\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)
| x-2 | 1 | -1 | 2 | -2 |
| x | 3 | 1 | 4 | 0 |
Vậy ...
đkcđ: x khác 0 và -3
\(A=\frac{x-3}{x}-\frac{x}{x-3}+\frac{9}{x.\left(x-3\right)}\)
\(A=\frac{\left(x-3\right)^2}{x.\left(x-3\right)}-\frac{x^2}{x.\left(x-3\right)}+\frac{9}{x.\left(x-3\right)}\)
\(A=\frac{x^2-6x+9-x^2+9}{x.\left(x-3\right)}=\frac{-6x+18}{x.\left(x-3\right)}=\frac{-6.\left(x-3\right)}{x.\left(x-3\right)}=-\frac{6}{x}\)
để A thuộc Z => 6 chia hết cho x
=>....
\(Taco\)
\(ĐKXD:x\ne0;x\ne3\)
\(\frac{x-3}{x}-\frac{x}{x-3}+\frac{9}{x^2-3x}=\frac{x-3}{x}-\frac{x}{x-3}+\frac{9}{x\left(x-3\right)}\)
\(=\frac{x^2-6x+9}{x\left(x-3\right)}-\frac{x^2}{x\left(x-3\right)}+\frac{9}{x\left(x-3\right)}=\frac{x^2-6x+9-x^2+9}{x\left(x-3\right)}\)
\(=\frac{18-6x}{x-3}\)
\(A\inℤ\Leftrightarrow18-6x⋮x-3\Leftrightarrow18-6x+6x-18⋮x-3\Leftrightarrow0⋮x-3\)
Vậy vs mọi GT của x thì A nguyên
a) ĐKXĐ của phương trình P là: \(\left[{}\begin{matrix}x\ne-3\\x\ne2\end{matrix}\right.\)
b) \(P=\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{2-x}\) (1)
Khi đó (1) \(\Leftrightarrow\) \(P\) \(=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+3\right)\left(x-2\right)}-\frac{5}{\left(x+3\right)\left(x-2\right)}-\frac{x+3}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{\left(x+2\right)\left(x-2\right)-5-\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}\) \(=\frac{x^2-4-5-x-3}{\left(x+3\right)\left(x-2\right)}\)
\(=\frac{x^2-x-12}{\left(x+3\right)\left(x-2\right)}\) \(=\frac{\left(x-4\right)\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}=\frac{x-4}{x-2}\) (Vì \(x+3\ne0\))
Mà \(P=\frac{-3}{4}\) nên \(\frac{x-4}{x-2}=-\frac{3}{4}\)
\(\Leftrightarrow4\left(x-4\right)=-3\left(x-2\right)\)
\(\Leftrightarrow4x-16=6-3x\)
\(\Leftrightarrow4x+3x=6+16\)
\(\Leftrightarrow7x=22\) \(\Leftrightarrow x=\frac{22}{7}\) (thỏa mãn ĐKXĐ)
Vậy để \(P=-\frac{3}{4}\) thì \(x=\frac{22}{7}\)
c) Để \(P\) nguyên thì \(\frac{x-4}{x-2}\in Z\) \(\Rightarrow x-4⋮x-2\)
\(\Leftrightarrow x-4-\left(x-2\right)⋮x-2\) (Vì \(\left(x-2\right)⋮x-2\))
\(\Leftrightarrow x-4-x+2⋮x-2\)
\(\Leftrightarrow-2⋮x-2\)
\(\Rightarrow x-2\in\)Ư(\(-2\))\(=\left\{1;-1;2;-2\right\}\)
\(\Leftrightarrow x\in\left\{3;1;4;0\right\}\)
Vậy để \(P\) nguyên thì \(x\in\left\{3;1;4;0\right\}\)
d) Ta có: \(x^2-9=0\Leftrightarrow x^2=9\Leftrightarrow x\in\left\{3;-3\right\}\)
Mà \(P=\frac{x-4}{x-2}\) (2) . Thay \(x=3\) vào (2) ta được:
\(P=\) \(\frac{3-4}{3-2}\) \(=-1\)
Thay \(x=-3\) vào (2) ta được :
\(P=\frac{-3-4}{-3-2}=\frac{7}{5}\)
Vậy giá trị của biểu thức \(P\) khi : \(x=3\) là \(-1\)
: \(x=-3\) là \(\frac{7}{5}\)