Chứng minh đẳng thức :
(\(\frac{1}{x^2+2x+1}\left(\frac{1}{x^2}+1\right)+\frac{2}{x^3+2x^2+x}\) ) : \(\frac{1}{4x^2-x^3}\) = 4 - x
Những câu hỏi liên quan
1.Tính:x:frac{x-1}{2}-frac{left(x-1right)left(x^2+4x+1right)}{2x^2+2x}.frac{-4x}{left(x-1right)^2}-frac{4x^2}{x^2-1}2.Chứng minh đẳng thức sau( giả sử đẳng thức có nghĩa):frac{y-z}{left(x-yright)left(x-zright)}+frac{z-x}{left(y-zright)left(y-xright)}+frac{x-y}{left(z-xright)left(z-yright)}frac{2}{x-y}+frac{2}{y-z}+frac{2}{z-x} Các bạn giúp mình với!
Đọc tiếp
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!
chứng minh đẳng thức
\(\left(\frac{x-y}{2y-x}-\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\right):\frac{x^4+4x^2y^2+y^4-4}{x^2+y+xy+x}:\frac{1}{2x^2+y+2}=\frac{x+1}{2y-x}\)
Rút gọn biểu thức sau: A=\(\left[\left(x^4-x+\frac{x-3}{x^3+1}\right).\frac{\left(x^3-2x^2+2x-1\right)\left(x+1\right)}{x^9+x^7-3x^2-3}+1-\frac{2\left(x+6\right)}{x^2+1}\right].\frac{4x^2+4x+1}{\left(x+4\right)\left(3-x\right)}\)
Chứng minh đẳng thức :
a)\(\left(\frac{9}{x^3-9x}+\frac{1}{x+3}\right):\left(\frac{x-3}{x^2+3x}-\frac{x}{3x+9}\right)=\frac{3}{3-x}\)
b) \([\frac{2}{3x}-\frac{2}{x+1}.\left(\frac{x+1}{3x}-x-1\right)]:\frac{x-1}{x}=\frac{2x}{x-1}\)
b) \(\left[\frac{2}{3x}-\frac{2}{x+1}.\left(\frac{x+1}{3x}-x-1\right)\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-\frac{2}{x+1}.\left(\frac{x+1}{3x}-\left(x+1\right)\right)\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-\frac{2}{x+1}.\left(x+1\right)\left(\frac{1}{3x}-1\right)\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-2\left(\frac{1}{3x}-1\right)\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-\frac{2}{3x}+2\right]:\frac{x-1}{x}\)
\(=2.\frac{x}{x-1}=\frac{2x}{x-1}\left(đpcm\right)\)
a) \(\left(\frac{9}{x^3-9x}+\frac{1}{x+3}\right):\left(\frac{x-3}{x^2+3x}-\frac{x}{3x+9}\right)\)
\(=\left(\frac{9}{x\left(x^2-9\right)}+\frac{1}{x+3}\right):\left(\frac{x-3}{x\left(x+3\right)}-\frac{x}{3\left(x+3\right)}\right)\)
\(=\left(\frac{9}{x\left(x+3\right)\left(x-3\right)}+\frac{x^2-3x}{x\left(x+3\right)\left(x-3\right)}\right)\)
\(:\left(\frac{3x-9}{3x\left(x+3\right)}-\frac{x^2}{3x\left(x+3\right)}\right)\)
\(=\frac{x^2-3x+9}{x\left(x+3\right)\left(x-3\right)}:\frac{-x^2+3x-9}{3x\left(x+3\right)}\)
\(=\frac{x^2-3x+9}{x\left(x+3\right)\left(x-3\right)}.\frac{3x\left(x+3\right)}{-x^2+3x-9}\)
\(=\frac{x^2-3x+9}{x-3}.\frac{3}{x^2+3x-9}\)
\(=\frac{x^2-3x+9}{3-x}.\frac{3}{x^2-3x+9}\)
\(=\frac{3}{3-x}\left(đpcm\right)\)
Bài 1: Thực hiện phép tính:a) frac{4x-4}{x^2-4x-4}:frac{x^2-1}{left(2-xright)^2}b) frac{2x+1}{2x^2-x}+frac{32x^2}{1-4x^2}+frac{1-2x}{2x^2+x} c) left(frac{1}{x+1}+frac{1}{x-1}-frac{2x}{1-x^2}right).frac{x-1}{4x}Bài 2: 1. Tìm n để đa thức x4 - x3 + 6x2 - x + n chia hết cho đa thức x2 - x + 52. Tìm n để đa thức 3x3 + 10x2 - 5 + n chia hết cho đa thức 3x + 1Bài 3:Cho biểu thức: N ( 4x + 3 )2 - 2x ( x + 6 ) - 5 ( x - 2 ) ( x + 2 ) Chứng minh biểu thức n luôn dương.
Đọc tiếp
Bài 1: Thực hiện phép tính:
a) \(\frac{4x-4}{x^2-4x-4}:\frac{x^2-1}{\left(2-x\right)^2}\)
b) \(\frac{2x+1}{2x^2-x}+\frac{32x^2}{1-4x^2}+\frac{1-2x}{2x^2+x}\)
c) \(\left(\frac{1}{x+1}+\frac{1}{x-1}-\frac{2x}{1-x^2}\right).\frac{x-1}{4x}\)
Bài 2:
1. Tìm n để đa thức x4 - x3 + 6x2 - x + n chia hết cho đa thức x2 - x + 5
2. Tìm n để đa thức 3x3 + 10x2 - 5 + n chia hết cho đa thức 3x + 1
Bài 3:
Cho biểu thức: N = ( 4x + 3 )2 - 2x ( x + 6 ) - 5 ( x - 2 ) ( x + 2 )
Chứng minh biểu thức n luôn dương.
Bài 1.
a)\(\frac{4x-4}{x^2-4x+4}\div\frac{x^2-1}{\left(2-x\right)^2}=\frac{4\left(x-1\right)}{\left(x-2\right)^2}\div\frac{\left(x-1\right)\left(x+1\right)}{\left(x-2\right)^2}=\frac{4\left(x-1\right)}{\left(x-2\right)^2}\times\frac{\left(x-2\right)^2}{\left(x-1\right)\left(x+1\right)}=\frac{4}{x+1}\)
b) \(\frac{2x+1}{2x^2-x}+\frac{32x^2}{1-4x^2}+\frac{1-2x}{2x^2+x}=\frac{2x+1}{x\left(2x-1\right)}+\frac{-32x^2}{4x^2-1}+\frac{1-2x}{x\left(2x+1\right)}\)
\(=\frac{\left(2x+1\right)\left(2x+1\right)}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-32x^3}{x\left(2x-1\right)\left(2x+1\right)}+\frac{\left(1-2x\right)\left(2x-1\right)}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{4x^2+4x+1}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-32x^3}{x\left(2x-1\right)\left(2x+1\right)}+\frac{-4x^2+4x-1}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{4x^2+4x+1-32x^3-4x^2+4x-1}{x\left(2x-1\right)\left(2x+1\right)}=\frac{-32x^3+8x}{x\left(2x-1\right)\left(2x+1\right)}\)
\(=\frac{-8x\left(4x^2-1\right)}{x\left(2x-1\right)\left(2x+1\right)}=\frac{-8x\left(2x-1\right)\left(2x+1\right)}{x\left(2x-1\right)\left(2x+1\right)}=-8\)
c) \(\left(\frac{1}{x+1}+\frac{1}{x-1}-\frac{2x}{1-x^2}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{1}{x+1}+\frac{1}{x-1}+\frac{2x}{x^2-1}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{x-1}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{2x}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x-1}{4x}\)
\(=\left(\frac{x-1+x+1+2x}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x-1}{4x}\)
\(=\frac{4x}{\left(x-1\right)\left(x+1\right)}\times\frac{x-1}{4x}=\frac{1}{x+1}\)
Bài 3.
N = ( 4x + 3 )2 - 2x( x + 6 ) - 5( x - 2 )( x + 2 )
= 16x2 + 24x + 9 - 2x2 - 12x - 5( x2 - 4 )
= 14x2 + 12x + 9 - 5x2 + 20
= 9x2 + 12x + 29
= 9( x2 + 4/3x + 4/9 ) + 25
= 9( x + 2/3 )2 + 25 ≥ 25 > 0 ∀ x
=> đpcm
Chứng minh đẳng thức:
\(\left[\frac{2}{3x}-\frac{2}{x+1}\left(\frac{x+1}{3x}-x-1\right)\right]:\frac{x-1}{x}=\frac{2x}{x-1}\)
\(\left[\frac{2}{3x}-\frac{2}{x+1}\left(\frac{x+1}{3x}-x-1\right)\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-\frac{2\left(x+1\right)}{\left(x+1\right).3x}-\frac{2\left(-x-1\right)}{x+1}\right]:\frac{x-1}{x}\)
\(=\)\(\left[\frac{2}{3x}-\frac{2\left(x+1\right)}{\left(x+1\right).3x}+\frac{2\left(x+1\right)}{x+1}\right]:\frac{x-1}{x}\)
\(=\left[\frac{2}{3x}-\frac{2}{3x}+2\right]:\frac{x-1}{x}\)
\(=2.\frac{x}{x-1}=\frac{2x}{x-1}\)\(\left(đpcm\right)\)
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\(P=\left[\left(x^4-x+\frac{x-3}{x^3+1}\right)\cdot\left(\frac{\left(x^3-2x^2-2x-1\right)\cdot\left(x+1\right)}{x^9+x^7-3x^2-3}\right)+1-\frac{2\left(x+6\right)}{x^2+1}\right]\cdot\frac{4x^2+4x+1}{\left(x+3\right)\left(4-x\right)}\)
a, Tìm ĐKXD của P
b,Rút Gọn P
c,Chứng Minh Với các giá trị của x mà biểu thức P có nghĩa thì \(-5\le P\le0\)
A,left(frac{x+1}{x-1}-frac{x-1}{x+1}right):left(frac{1}{x+1}-frac{x}{1-x}+frac{2}{x^2-1}right)frac{4x}{left(x+1right)^2}B,frac{2+x}{2-x}:frac{4x^2}{4-4x+x^2}cdotleft(frac{2}{2-x}-frac{4}{8+x^2}cdotfrac{4-2x+x^2}{2-x}right)frac{1}{2x}C,left[left(frac{3}{x-y}+frac{3x}{x^2-y^2}right):frac{2x+y}{x^2+2xy+y^2}right]cdotfrac{x-y}{3}xyChứng minh đẳng thức ( tìm x)mọi người giải dùm mình cảm ơn
Đọc tiếp
\(A,\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right):\left(\frac{1}{x+1}-\frac{x}{1-x}+\frac{2}{x^2-1}\right)=\frac{4x}{\left(x+1\right)^2}\)
\(B,\frac{2+x}{2-x}:\frac{4x^2}{4-4x+x^2}\cdot\left(\frac{2}{2-x}-\frac{4}{8+x^2}\cdot\frac{4-2x+x^2}{2-x}\right)=\frac{1}{2x}\)
\(C,\left[\left(\frac{3}{x-y}+\frac{3x}{x^2-y^2}\right):\frac{2x+y}{x^2+2xy+y^2}\right]\cdot\frac{x-y}{3}=xy\)
Chứng minh đẳng thức ( tìm x)
mọi người giải dùm mình cảm ơn
a VT=.\(\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right):\left(\frac{1}{x+1}-\frac{x}{1-x}+\frac{2}{x^2-1}\right)\)
=\(\frac{\left(x+1\right)^2-\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)}:\frac{x-1+x\left(x-1\right)+2}{\left(x+1\right)\left(x-1\right)}\)
\(=\frac{x^2+2x+1-x^2+2x-1}{\left(x+1\right)\left(x-1\right)}.\frac{\left(x+1\right)\left(x-1\right)}{x^2+2x+1}\)
\(=\frac{4x}{\left(x+1\right)^2}\)=VP
b.VT\(=\frac{2+x}{2-x}.\frac{\left(2-x\right)^2}{4x^2}.\left(\frac{2}{2-x}-\frac{4}{\left(x+2\right)\left(x^2-2x+4\right)}.\frac{4-2x+x^2}{2-x}\right)\)
=\(\frac{4-x^2}{4x^2}.\left(\frac{2}{2-x}-\frac{4}{4-x^2}\right)=\frac{4-x^2}{4x^2}.\frac{2\left(2+x\right)-4}{4-x^2}\)
=\(\frac{2x}{4x^2}=\frac{1}{2x}\)=VP
c VT=.\(\left[\left(\frac{3}{x-y}+\frac{3x}{x^2-y^2}\right).\frac{\left(x+y\right)^2}{2x+y}\right].\frac{x-y}{3}\)
\(=\left[\frac{3\left(x+y\right)+3x}{\left(x+y\right)\left(x-y\right)}.\frac{\left(x+y\right)^2}{2x+y}\right].\frac{x-y}{3}\)
\(=\frac{3\left(2x+y\right)\left(x+y\right)^2}{\left(x+y\right)\left(x-y\right)\left(2x+y\right)}.\frac{x-y}{3}\)
\(=x+y=\)VP
Vậy các đẳng thức được chứng minh
=
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Chứng minh bất đẳng thức:
\(\left[\frac{2}{3x}-\frac{2}{x+1}.\left(\frac{x+1}{3x}-x-1\right)\right]:\frac{x-1}{x}=\frac{2x}{x-1}\)