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a) Đk: x > 0 và x khác +-1
Ta có: A = \(\left(\frac{x+1}{x}-\frac{1}{1-x}-\frac{x^2-2}{x^2-x}\right):\frac{x^2+x}{x^2-2x+1}\)
A = \(\left[\frac{\left(x-1\right)\left(x+1\right)+x-x^2+2}{x\left(x-1\right)}\right]:\frac{x\left(x+1\right)}{\left(x-1\right)^2}\)
A = \(\frac{x^2-1+x-x^2+2}{x\left(x-1\right)}\cdot\frac{\left(x-1\right)^2}{x\left(x+1\right)}\)
A = \(\frac{x+1}{x}\cdot\frac{x-1}{x\left(x+1\right)}=\frac{x-1}{x^2}\)
b) Ta có: A = \(\frac{x-1}{x^2}=\frac{1}{x}-\frac{1}{x^2}=-\left(\frac{1}{x^2}-\frac{1}{x}+\frac{1}{4}\right)+\frac{1}{4}=-\left(\frac{1}{x}-\frac{1}{2}\right)^2+\frac{1}{4}\le\frac{1}{4}\forall x\)
Dấu "=" xảy ra <=> 1/x - 1/2 = 0 <=> x = 2 (tm)
Vậy MaxA = 1/4 <=> x = 2
a, ĐKXĐ: \(\hept{\begin{cases}x^3+1\ne0\\x^9+x^7-3x^2-3\ne0\\x^2+1\ne0\end{cases}}\)
b, \(Q=\left[\left(x^4-x+\frac{x-3}{x^3+1}\right).\frac{\left(x^3-2x^2+2x-1\right)\left(x+1\right)}{x^9+x^7-3x^2-3}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\left[\frac{\left(x^3+1\right)\left(x^4-x\right)+x-3}{\left(x+1\right)\left(x^2-x+1\right)}.\frac{\left(x-1\right)\left(x+1\right)\left(x^2-x+1\right)}{\left(x^7-3\right)\left(x^2+1\right)}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\left[\left(x^7-3\right).\frac{\left(x-1\right)}{\left(x^7-3\right)\left(x^2+1\right)}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\frac{x-1+x^2+1-2x-12}{x^2+1}\)
\(Q=\frac{\left(x-4\right)\left(x+3\right)}{x^2+1}\)
a
\(ĐKXĐ:x\ne3;x\ne-3;x\ne0\)
b
\(A=\left(\frac{9}{x^3-9x}+\frac{1}{x+3}\right):\left(\frac{x-3}{x^2+3x}-\frac{x}{3x+9}\right)\)
\(=\left[\frac{9}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x+3}\right]:\left[\frac{x-3}{x\left(x+3\right)}-\frac{x}{3\left(x+3\right)}\right]\)
\(=\frac{9+x^2-3x}{x\left(x-3\right)\left(x+3\right)}:\frac{3x-9-x^2}{3x\left(x+3\right)}\)
\(=\frac{9+x^2-3x}{x\left(x-3\right)\left(x+3\right)}\cdot\frac{3x\left(x+3\right)}{-\left(9-3x+x^2\right)}=\frac{-3}{x-3}\)
c
Với \(x=4\Rightarrow A=-3\)
d
Để A nguyên thì \(\frac{3}{x-3}\) nguyên
\(\Rightarrow3⋮x-3\)
Làm nốt.
\(a,x\ne2;x\ne-2;x\ne0\)
\(b,A=\left(\frac{x}{x^2-4}+\frac{2}{2-x}+\frac{1}{x+2}\right):\frac{6}{x+2}\)
\(=\frac{x-2\left(x+2\right)+x-2}{\left(x-2\right)\left(x+2\right)}:\frac{6}{x+2}\)
\(=\frac{-6}{\left(x-2\right)\left(x+2\right)}:\frac{6}{x+2}\)
\(=\frac{-6}{\left(x-2\right)\left(x+2\right)}.\frac{x+2}{6}\)
\(=\frac{1}{2-x}\)
\(c,\)Để A > 0 thi \(\frac{1}{2-x}>0\Leftrightarrow2-x>0\Leftrightarrow x< 2\)
bài1 A=\(\left(\frac{3-x}{x+3}\cdot\frac{x^2+6x+9}{x^2-9}+\frac{x}{x+3}\right):\frac{3x^2}{x+3}\)
=\(\left(-\frac{x-3\cdot\left(x+3\right)^2}{\left(x+3\right)^2\cdot\left(x-3\right)}+\frac{x}{x+3}\right):\frac{3x^2}{x+3}\)
=\(-\frac{x}{x+3}\cdot\frac{x+3}{3x^2}=\frac{-1}{3x}\)
b) thế \(x=-\frac{1}{2}\)vào biểu thức A
\(-\frac{1}{3\cdot\left(-\frac{1}{2}\right)}=\frac{2}{3}\)
c) A=\(-\frac{1}{3x}< 0\)
VÌ (-1) <0 nên 3x>0
x >0
a, ĐKXĐ: \(x\ne-3\) và \(x\ne\pm1\)
b, \(P=\frac{x\left(x+3\right)-11+x^2-3x+9}{x^3+27}:\frac{x^2-1}{x+3}\)
\(P=\frac{2x^2-2}{x^3+27}.\frac{x+3}{x^2-1}\)
\(=\frac{2\left(x-1\right)\left(x+1\right)}{\left(x+3\right)\left(x^2-3x+9\right)}.\frac{x+3}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{2}{x^2-3x+9}\)
c, \(P=\frac{2}{x^2-3x+9}==\frac{2}{\left(x-\frac{3}{2}\right)^2+\frac{27}{4}}\le\frac{2}{\frac{27}{4}}=\frac{8}{27}\)
Dấu "=" xảy ra khi: \(x-\frac{3}{2}=0\Leftrightarrow x=\frac{3}{2}\)
Vậy P lớn nhất bằng \(\frac{8}{27}\) \(\Leftrightarrow x=\frac{3}{2}\)
\(P=\left(\frac{x}{x^2-3x+9}-\frac{11}{x^3+27}+\frac{1}{x+3}\right):\frac{x^2-1}{x+3}.\)
ĐKXĐ : \(x\ne-3;x\ne0\)
\(P=\left(\frac{x\left(x+3\right)}{\left(x+3\right)\left(x^2-3x+9\right)}-\frac{11}{\left(x+3\right)\left(x^2-3x+9\right)}+\frac{x^2-3x+9}{\left(x+3\right)\left(x^2-3x+9\right)}\right).\frac{x+3}{x^2-1}\)
\(P=\left(\frac{x^2+3x-11+x^2-3x+9}{\left(x+3\right)\left(x^2-3x+9\right)}\right).\frac{x+3}{x^2-1}\)
\(P=\frac{2x^2-2}{\left(x^2-3x+9\right)}.\frac{1}{x^2-1}=\frac{2\left(x^2-1\right)}{\left(x^2-3x+9\right)}.\frac{1}{x^2-1}\)
\(P=\frac{2}{x^2-3x+9}\)
a) Điều kiện: \(x\ne0;x\ne1\)
b) \(A=\left(\frac{x}{x-1}-\frac{1}{x^2-x}\right):\frac{x^2+2x+1}{x}\)
\(A=\left(\frac{x}{x-1}-\frac{1}{x.\left(x-1\right)}\right):\frac{\left(x+1\right)^2}{x}\)
\(A=\left(\frac{x^2}{\left(x-1\right).x}-\frac{1}{x.\left(x-1\right)}\right):\frac{\left(x+1\right)^2}{x}\)
\(A=\frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right).x}.\frac{x}{\left(x+1\right)^2}\)
\(A=\frac{x+1}{x}.\frac{x}{\left(x+1\right)^2}=\frac{1}{x+1}\)
c) Thay: \(x=2\)vào \(\frac{1}{x+1}\)ta có: \(A=\frac{1}{2+1}=\frac{1}{3}\)
a) ĐKXĐ : \(\hept{\begin{cases}x\ne0\\x\ne1\end{cases}}\)
b)
\(A=\left(\frac{x}{x-1}-\frac{1}{x^2-x}\right):\frac{x^2+2x+1}{x}\)
\(A=\left(\frac{x}{x-1}-\frac{1}{x\left(x-1\right)}\right)\cdot\frac{x}{x^2+2x+1}\)
\(A=\left(\frac{x\cdot x}{x\left(x-1\right)}-\frac{1}{x\left(x-1\right)}\right)\cdot\frac{x}{\left(x+1\right)^2}\)
\(A=\frac{x^2-1}{x\left(x-1\right)}\cdot\frac{x}{\left(x+1\right)^2}=\frac{\left(x^2-1\right)\cdot x}{x\left(x-1\right)\left(x+1\right)^2}=\frac{\left(x+1\right)\left(x-1\right)\cdot x}{x\left(x-1\right)\left(x+1\right)^2}=\frac{1}{x+1}\)
c) \(A=\frac{1}{x+1}=\frac{1}{2+1}=\frac{1}{3}\)
Vậy \(A=\frac{1}{3}\)
Câu 3 :
\(a,A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right):\frac{2x}{5x-5}\) ĐKXđ : \(x\ne\pm1\)
\(A=\left(\frac{\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}-\frac{\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)}\right):\frac{2x}{5\left(x-1\right)}\)
\(A=\left(\frac{x^2+2x+1-x^2+2x-1}{\left(x-1\right)\left(x+1\right)}\right).\frac{5\left(x-1\right)}{2x}\)
\(A=\frac{4x}{\left(x-1\right)\left(x+1\right)}.\frac{5\left(x-1\right)}{2x}\)
\(A=\frac{10}{x+1}\)
\(B=\left(\frac{x}{3x-9}+\frac{2x-3}{3x-x^2}\right).\frac{3x^2-9x}{x^2-6x+9}.\)
ĐKXđ : \(x\ne0;x\ne3\)
\(B=\left(\frac{x}{3\left(x-3\right)}+\frac{2x-3}{x\left(3-x\right)}\right).\frac{3x\left(x-3\right)}{x^2-6x+9}\)
\(B=\left(\frac{x^2}{3x\left(x-3\right)}+\frac{9-6x}{3x\left(x-3\right)}\right).\frac{3x\left(x-3\right)}{x^2-6x+9}\)
\(B=\frac{x^2-6x+9}{3x\left(x-3\right)}.\frac{3x\left(x-3\right)}{x^2-6x+9}=1\)