Rút gọn : \(\left(\frac{1}{2x-y}+\frac{3y}{x^2-4x^2}-\frac{2}{2x+y}\right):\left(\frac{4x^2+y^2}{4x^2-y^2}+1\right)\)
Rút gọn : \(\left(\frac{1+x}{x}+\frac{1}{4x^2}\right)\left(\frac{1-2x}{1+2x}-\frac{1}{1-4x^2}\times\frac{1-4x+4x^2}{1+2x}\right)-\frac{1}{2x}\)
\(=\dfrac{4x\left(x+1\right)+1}{4x^2}\cdot\left(\dfrac{-\left(2x-1\right)}{2x+1}+\dfrac{1}{\left(2x+1\right)\left(2x-1\right)}\cdot\dfrac{\left(2x-1\right)^2}{2x+1}\right)-\dfrac{1}{2x}\)
\(=\dfrac{\left(2x+1\right)^2}{4x^2}\cdot\left(\dfrac{-\left(2x-1\right)}{2x+1}+\dfrac{2x-1}{\left(2x+1\right)^2}\right)-\dfrac{1}{2x}\)
\(=\dfrac{\left(2x+1\right)^2}{4x^2}\cdot\dfrac{-\left(2x-1\right)\left(2x+1\right)+2x-1}{\left(2x+1\right)^2}-\dfrac{1}{2x}\)
\(=\dfrac{-4x^2+1+2x-1}{4x^2}-\dfrac{1}{2x}\)
\(=\dfrac{-4x^2+2x}{4x^2}-\dfrac{1}{2x}\)
\(=\dfrac{-2x\left(2x-1\right)}{2x\cdot2x}-\dfrac{1}{2x}\)
\(=\dfrac{-2x+1-1}{2x}=\dfrac{-2x}{2x}=-1\)
Tính
a, \(\frac{1}{\left(y-1\right)\left(y-2\right)}+\frac{2}{\left(2-y\right)\left(3y-y\right)}+\frac{3}{\left(1-y\right)\left(y-3\right)}\)
b, \(\frac{x^2}{x+1}+\frac{2x}{x^2-1}-\frac{1}{1-x}+1\)
c, \(\frac{1}{x^2+3x+2}-\frac{2x}{x^2+4x^2+4x}+\frac{1}{x^2+5x+6}\)
Rút gọn: \(\left(\frac{x-y}{2y-x}-\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\right):\frac{4x^4+4x^2y+y^2-4}{x^2+xy+x+y}:\frac{x+1}{2y^2+y+2}\)
Rút gọn A : \(\left[\frac{\left(x-1\right)^2}{3x+\left(x-1\right)^2}-\frac{1-2x^2+4x}{x^3-1}+\frac{1}{x-1}\right]:\frac{2x}{x^3+x}\)
\(A=\left(\dfrac{x^2-2x+1}{x^2+x+1}-\dfrac{-2x^2+4x+1}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{x-1}\right):\dfrac{2x}{x^3+x}\)
\(=\dfrac{x^3-3x^2+3x-1+2x^2-4x-1+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\cdot\dfrac{x^2+1}{2}\)
\(=\dfrac{x^3-1}{\left(x-1\right)\left(x^2+x+1\right)}\cdot\dfrac{x^2+1}{2}=\dfrac{x^2+1}{2}\)
\(\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16x}\)
\(\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16x}\)
\(=\frac{\left(2x+y\right)^22\left(4x^2-y^2\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\left(2x+y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{16x^2}{16x\left(2x-y\right)^2}=\frac{x}{\left(2x-y\right)^2}\)
\(\left[\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-4^2}+\frac{1}{\left(2x+y\right)^2}\right].\frac{4x^2+4xy+y^2}{16x}\)
\(=\frac{\left(2x+y\right)^22\left(4x^2-y^2\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\left(2x+y\right)^2}.\frac{\left(2x+y\right)^2}{16x}\)
\(=\frac{16x^2}{16x\left(2x-y\right)^2}=\frac{x}{\left(2x-y\right)^2}\)
1.Tính:
\(x:\frac{x-1}{2}-\frac{\left(x-1\right)\left(x^2+4x+1\right)}{2x^2+2x}.\frac{-4x}{\left(x-1\right)^2}-\frac{4x^2}{x^2-1}\)
2.Chứng minh đẳng thức sau( giả sử đẳng thức có nghĩa):
\(\frac{y-z}{\left(x-y\right)\left(x-z\right)}+\frac{z-x}{\left(y-z\right)\left(y-x\right)}+\frac{x-y}{\left(z-x\right)\left(z-y\right)}=\frac{2}{x-y}+\frac{2}{y-z}+\frac{2}{z-x}\)
Các bạn giúp mình với!
Rút gọn: \(\left(\frac{3x}{1-2x}+\frac{2x}{1+2x}\right):\frac{2x^2+5x}{1-4x+4x^2}\)
\(=\dfrac{3x+6x^2+2x-4x^2}{\left(1-2x\right)\left(1+2x\right)}\cdot\dfrac{\left(1-2x\right)^2}{x\left(2x+5\right)}\)
\(=\dfrac{1-2x}{1+2x}\)
\(\frac{4xy}{y^2-x^2}:\left(\frac{1}{x^2+2xy+y^2}-\frac{1}{x^2-y^2}\right)\)
\(\left(\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right).\frac{4x^2+4xy+y^2}{16x}\)
Rút gọn các biểu thức rồi tính giá trị:
a) \(\frac{x^2y\left(y-x\right)-xy^2\left(x-y\right)}{3y^2-2x^2}\), với x = -3; y = \(\frac{1}{2}\)
b) \(\frac{\left(8x^3-y^3\right)\left(4x^2-y^2\right)}{\left(2x+y\right)\left(4x^2-4xy+y^2\right)}\), với x = 2; y = -\(\frac{1}{2}\)
Lời giải:
a)
\(A=\frac{x^2y(y-x)-xy^2(x-y)}{3y^2-2x^2}=\frac{x^2y(y-x)+xy^2(y-x)}{3y^2-2x^2}=\frac{(xy^2+x^2y)(y-x)}{3y^2-2x^2}\)
\(=\frac{xy(x+y)(y-x)}{3y^2-2x^2}=\frac{xy(y^2-x^2)}{3y^2-2x^2}\)
Với $x=-3; y=\frac{1}{2}$ thì:
$xy=\frac{-3}{2}; x^2=9; y^2=\frac{1}{4}$
Do đó $A=\frac{-35}{46}$
b)
\(B=\frac{(8x^3-y^3)(4x^2-y^2)}{(2x+y)(4x^2-4xy+y^2)}=\frac{(2x-y)(4x^2+2xy+y^2)(2x-y)(2x+y)}{(2x+y)(2x-y)^2}\)
\(=4x^2+2xy+y^2=4.2^2+2.2.\frac{-1}{2}+(\frac{-1}{2})^2=\frac{57}{4}\)