Cho A=\(\frac{x^3+26x-19}{x^2+2x-3}-\frac{2x}{x-1}+\frac{x-3}{x+3}\)
a. Rút gọn A
b. Tìm x để A=4
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14 tháng 7 2017
ĐK \(\hept{\begin{cases}x\ne1\\x\ne-3\end{cases}}\)
a, \(P=\frac{x^2+26x-19}{\left(x+3\right)\left(x-1\right)}-\frac{2x}{x-1}+\frac{x-3}{x+3}\)\(=\frac{x^2+26x-19-2x\left(x+3\right)+\left(x-3\right)\left(x-1\right)}{\left(x+3\right)\left(x-1\right)}\)
\(=\frac{x^2+26x-19-2x^2-6x+x^2-4x+3}{\left(x+3\right)\left(x-1\right)}\)\(=\frac{16\left(x-1\right)}{\left(x+3\right)\left(x-1\right)}=\frac{16}{x+3}\)
b. Với \(x=3\Rightarrow P=\frac{16}{3+3}=\frac{8}{3}\)
Với \(x=-1\Rightarrow P=\frac{16}{-1+3}=8\)
c. \(P=4\Rightarrow\frac{16}{x+3}=4\Rightarrow x+3=4\Rightarrow x=1\)
d. \(P\in Z\Rightarrow x+3\inƯ\left(16\right)\)
\(\Rightarrow x+3\in\left\{-16;-8;-4;-2;-1;1;2;4;8;16\right\}\)
\(\Rightarrow x\in\left\{-19;-11;-7;-5;-4;-2;-1;1;5;13\right\}\)
NN
Nguyễn Ngọc Anh Minh
CTVHS
VIP
14 tháng 7 2017
\(P=\frac{x^2+26x-19}{\left(x-1\right)\left(x+3\right)}-\frac{2x}{x-1}+\frac{x-3}{x+3}=\)
\(P=\frac{x^2+26x-19-2x\left(x+3\right)+\left(x-3\right)\left(x-1\right)}{\left(x-1\right)\left(x+3\right)}=\)
\(P=\frac{x^2+26x-19-2x^2-6x+x^2-4x+4}{\left(x-1\right)\left(x+3\right)}=\)
\(P=\frac{16x-15}{\left(x-1\right)\left(x+3\right)}\)
24 tháng 6 2017
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
18 tháng 1 2021
\(A=\frac{2x-9}{x^2-5x+6}-\frac{x+3}{x-2}-\frac{2x+4}{3-x}\)
a) ĐKXĐ : \(\hept{\begin{cases}x\ne2\\x\ne3\end{cases}}\)
\(A=\frac{2x-9}{\left(x-2\right)\left(x-3\right)}-\frac{x+3}{x-2}+\frac{2x+4}{x-3}\)
\(=\frac{2x-9}{\left(x-2\right)\left(x-3\right)}-\frac{\left(x+3\right)\left(x-3\right)}{\left(x-2\right)\left(x-3\right)}+\frac{\left(2x+4\right)\left(x-2\right)}{\left(x-2\right)\left(x-3\right)}\)
\(=\frac{2x-9}{\left(x-2\right)\left(x-3\right)}-\frac{x^2-9}{\left(x-2\right)\left(x-3\right)}+\frac{2x^2-8}{\left(x-2\right)\left(x-3\right)}\)
\(=\frac{2x-9-x^2+9+2x^2-8}{\left(x-2\right)\left(x-3\right)}\)
\(=\frac{x^2+2x-8}{\left(x-2\right)\left(x-3\right)}=\frac{\left(x-2\right)\left(x+4\right)}{\left(x-2\right)\left(x-3\right)}=\frac{x+4}{x-3}\)
b) Ta có : \(A=\frac{x+4}{x-3}=\frac{x-3+7}{x-3}=1+\frac{7}{x-3}\)
Để A đạt giá trị nguyên thì \(\frac{7}{x-3}\)đạt giá trị nguyên
=> 7 ⋮ x - 3
=> x - 3 ∈ Ư(7) = { ±1 ; ±7 }
| x-3 | 1 | -1 | 7 | -7 |
| x | 4 | 2 | 10 | -4 |
So với ĐKXĐ ta thấy x = 4 , x = 10 , x = -4 thỏa mãn
Vậy với x ∈ { ±4 ; 10 } thì A đạt giá trị nguyên
18 tháng 1 2021
(....) dùng để nhìn được chữ số ở phân số cuối cùng thôi, ko dùng để làm gì.
( ác ) là từ ( các )
(gia strij) là từ ( giá trị )
B
8 tháng 12 2016
a) \(A=\frac{4x-1}{x-2}-\frac{x-3}{x-1}+\frac{-2x+4}{x^2-3x+2}\)
\(\Leftrightarrow A=\frac{4x-1}{x-2}-\frac{x-3}{x-1}+\frac{-2x+4}{x^2-x-2x+2}\)
\(\Leftrightarrow A=\frac{4x-1}{x-2}-\frac{x-3}{x-1}+\frac{-2x+4}{x\left(x-1\right)-2\left(x-1\right)}\)
\(\Leftrightarrow A=\frac{4x-1}{x-2}-\frac{x-3}{x-1}+\frac{-2x+4}{\left(x-1\right)\left(x-2\right)}\)
\(\Leftrightarrow A=\frac{\left(4x-1\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)-2x+4}{\left(x-2\right)\left(x-1\right)}\)
\(\Leftrightarrow A=\frac{4x^2-4x-x+1-x^2+2x+3x-6-2x+4}{\left(x-2\right)\left(x-1\right)}\)
\(\Leftrightarrow A=\frac{3x^2-2x-1}{\left(x-2\right)\left(x-1\right)}\)
\(\Leftrightarrow A=\frac{3x^2-3x+\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}\)\(=\frac{3x\left(x-1\right)+\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}\)\(=\frac{\left(x-1\right)\left(3x+1\right)}{\left(x-2\right)\left(x-1\right)}\)\(=\frac{3x+1}{x-2}\)
b)\(\frac{3x+1}{x-2}=\frac{3x-6+7}{x-2}=\frac{3x-6}{x-2}+\frac{7}{x-2}=3+\frac{7}{x-2}\)
Ta có : \(x-2\inƯ_7\left\{-7;-1;1;7\right\}\)
\(\Rightarrow\left[\begin{array}{nghiempt}x-2=-7\\x-2=-1\\x-2=1\\x-2=7\end{array}\right.\)\(\Rightarrow\left[\begin{array}{nghiempt}\text{x=-5}\\\text{x=1}\\\text{x=3}\\\text{x}=9\end{array}\right.\)
\(\text{x}=1\) (loại)
Vậy giá trị nguyên tập hợp x là:
x=-5;3;9
7 tháng 8 2016
\(M=\frac{x^3+26x-19}{x^2+2x-3}-\frac{2x}{x-1}+\frac{x-3}{x+3}\)
\(=\frac{x^3+26x-19}{\left(x-1\right)\left(x+3\right)}-\frac{2x\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}+\frac{\left(x-1\right)\left(x-3\right)}{\left(x-1\right)\left(x+3\right)}\)
\(=\frac{x^3+26x-19-2x^2-6x+x^2-4x+3}{\left(x-1\right)\left(x+3\right)}\)
\(=\frac{x^3-x^2+16x-16}{\left(x-1\right)\left(x+3\right)}=\frac{x^2\left(x-1\right)+16\left(x-1\right)}{\left(x-1\right)\left(x+3\right)}\)
\(=\frac{\left(x^2+16\right)\left(x-1\right)}{\left(x-1\right)\left(x+3\right)}=\frac{x^2+16}{x+3}\)
DL
23 tháng 7 2019
a) \(B=\frac{1}{x+3}+\frac{x}{x-1}-\frac{4x}{x^2+2x-3}=\frac{x-1}{x^2+2x-3}+\frac{x^2+3x}{x^2+2x-3}-\frac{4x}{x^2+2x-3}\)
\(\Leftrightarrow B=\frac{x-1+x^2+3x-4x}{x^2+2x-3}=\frac{x^2-1}{x^2+2x+1-4}=\frac{\left(x-1\right)\left(x+1\right)}{\left(x+1\right)^2-2^2}\)
\(\Leftrightarrow B=\frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}=\frac{x+1}{x+3}\)
b) \(\frac{A-1}{B}=\frac{\frac{x-1}{x+3}-1}{\frac{x+1}{x+3}}=\frac{\frac{-4}{x+3}}{\frac{x+1}{x+3}}=\frac{-4}{x+1}\le\frac{1}{2}\Leftrightarrow-8\le x+1\Leftrightarrow x\ge-9\)