Chứng minh biểu thức không phụ thuộc vào biến:
A= (\(\dfrac{2}{2a-b}\)+\(\dfrac{6b}{b^{2^{ }}-4a^{2^{ }}}\)-\(\dfrac{4}{2a+b}\)):(1+\(\dfrac{4a^{2^{ }}+b^{2^{ }}}{4a^{2^{ }}-b^2}\))
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12 tháng 12 2021
\(M=a+\dfrac{4a+2ab+2b+b^2+4a-2ab-2b+b^2-4a}{\left(2-b\right)\left(2+b\right)}\\ M=a+\dfrac{4a+2b^2}{\left(2-b\right)\left(2+b\right)}=\dfrac{4a-ab^2+4a+2b^2}{\left(2-b\right)\left(2+b\right)}\\ M=\dfrac{8a-ab^2+2b^2}{4-b^2}\)
Ta có \(8a-b^2\left(a-2\right)=8a-\dfrac{a^2\left(a-2\right)}{\left(a+1\right)^2}=\dfrac{8a^3+16a^2+8a-a^3+2a^2}{\left(a+1\right)^2}=\dfrac{7a^3+18a^2+8a}{\left(a+1\right)^2}\)
\(4-b^2=4-\dfrac{a^2}{\left(a+1\right)^2}=\dfrac{4a^2+8a+4-a^2}{\left(a+1\right)^2}=\dfrac{3a^2+8a+4}{\left(a+1\right)^2}\)
\(\Leftrightarrow M=\dfrac{7a^3+18a^2+8a}{3a^2+8a+4}=\dfrac{a\left(7a+4\right)\left(a+2\right)}{\left(3a+2\right)\left(a+2\right)}=\dfrac{a\left(7a+4\right)}{3a+2}\)
21 tháng 5 2022
\(=\left(\dfrac{2}{2a-b}-\dfrac{6b}{\left(2a-b\right)\left(2a+b\right)}-\dfrac{4}{2a+b}\right):\dfrac{4a^2-b^2+4a^2+b^2}{4a^2-b^2}\)
\(=\dfrac{4a+2b-6b-8a+4b}{\left(2a-b\right)\left(2a+b\right)}\cdot\dfrac{\left(2a-b\right)\left(2a+b\right)}{8a^2}\)
\(=\dfrac{-4a}{8a^2}=\dfrac{-1}{2a}\)
DX
23 tháng 12 2022
2.
\(P=\left(\dfrac{a+6}{3\left(a+3\right)}-\dfrac{1}{a+3}\right).\dfrac{27a}{a+2}=\left(\dfrac{a+3}{3\left(a+3\right)}\right).\dfrac{27a}{a+2}=\dfrac{27a}{3\left(a+2\right)}=\dfrac{9a}{a+2}\)
ĐKXĐ là :
\(a\ne0;-3;-2\)
Vs a = 1 ta có:
=> P=3
1.
\(M=\left(\dfrac{2a}{2a+b}-\dfrac{4a^2}{\left(2a+b\right)^2}\right):\left(\dfrac{2a}{\left(2a-b\right)\left(2a+b\right)}-\dfrac{1}{2a-b}\right)=\left(\dfrac{4a^2+2ab-4a^2}{\left(2a+b\right)^2}\right).\left(\dfrac{\left(2a+b\right)\left(2a-b\right)}{b}\right)=\dfrac{2a.\left(2a-b\right)}{\left(2a+b\right)}\)
21 tháng 5 2022
\(=\left(\dfrac{2\left(2a+b\right)-6b-4\left(2a-b\right)}{\left(2a-b\right)\left(2a+b\right)}\right):\dfrac{4a^2-b^2+4a^2+b^2}{\left(2a-b\right)\left(2a+b\right)}\)
\(=\dfrac{4a+2b-6b-8a+4b}{8a^2}\)
\(=\dfrac{-4a}{8a^2}=\dfrac{-1}{2a}\)
5 tháng 6 2021
\(4.\left(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}-\dfrac{3}{2}\right)+\dfrac{ab^2+bc^2+ca^2+abc}{a^2b+b^2c+c^2a+abc}-1\ge0\)
\(\Leftrightarrow\dfrac{\left(a-b\right)\left(b-c\right)\left(c-a\right)}{a^2b+b^2c+c^2a+abc}-2.\dfrac{\left(a-b\right)\left(b-c\right)\left(c-a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge0\)
\(\Leftrightarrow\dfrac{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left[\left(a+b\right)\left(b+c\right)\left(c+a\right)-2\left(a^2b+b^2c+c^2a+abc\right)\right]}{\left(a^2b+b^2c+c^2a+abc\right)\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge0\)
\(\Leftrightarrow\dfrac{\left[\left(a-b\right)\left(b-c\right)\left(c-a\right)\right]^2}{\left(a^2b+b^2c+c^2a+abc\right)\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge0\)
Bất đẳng thức hiển nhiên đúng
Vậy ta có điều phải chúng minh. Dấu hằng đẳng thức xảy ra khi \(a=b=c\)
-Chúc bạn học tốt-
5 tháng 6 2021
Bạn giải thích hộ mình từ dòng 1 xuống dòng 2 đc ko ạ ?
11 tháng 8 2021
\(P=\dfrac{4a^2}{4b+2c}+\dfrac{4b^2}{4a+2c}+\dfrac{c^2}{4a+4b}\ge\dfrac{\left(2a+2b+c\right)^2}{8a+8b+4c}\)
\(=\dfrac{\left(2a+2b+c\right)^2}{4\left(2a+2b+c\right)}=\dfrac{1}{4}\left(2a+2b+c\right)\)
11 tháng 11 2023
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)
=>\(a=bk;c=dk\)
1: \(\dfrac{2a+3c}{2b+3d}=\dfrac{2\cdot bk+3\cdot dk}{2b+3d}=\dfrac{k\left(2b+3d\right)}{2b+3d}=k\)
\(\dfrac{2a-3c}{2b-3d}=\dfrac{2bk-3dk}{2b-3d}=\dfrac{k\left(2b-3d\right)}{2b-3d}=k\)
Do đó: \(\dfrac{2a+3c}{2b+3d}=\dfrac{2a-3c}{2b-3d}\)
2: \(\dfrac{4a-3b}{4c-3d}=\dfrac{4\cdot bk-3b}{4\cdot dk-3d}=\dfrac{b\left(4k-3\right)}{d\left(4k-3\right)}=\dfrac{b}{d}\)
\(\dfrac{4a+3b}{4c+3d}=\dfrac{4bk+3b}{4dk+3d}=\dfrac{b\left(4k+3\right)}{d\left(4k+3\right)}=\dfrac{b}{d}\)
Do đó: \(\dfrac{4a-3b}{4c-3d}=\dfrac{4a+3b}{4c+3d}\)
3: \(\dfrac{3a+5b}{3a-5b}=\dfrac{3bk+5b}{3bk-5b}=\dfrac{b\left(3k+5\right)}{b\left(3k-5\right)}=\dfrac{3k+5}{3k-5}\)
\(\dfrac{3c+5d}{3c-5d}=\dfrac{3dk+5d}{3dk-5d}=\dfrac{d\left(3k+5\right)}{d\left(3k-5\right)}=\dfrac{3k+5}{3k-5}\)
Do đó: \(\dfrac{3a+5b}{3a-5b}=\dfrac{3c+5d}{3c-5d}\)
4: \(\dfrac{3a-7b}{b}=\dfrac{3bk-7b}{b}=\dfrac{b\left(3k-7\right)}{b}=3k-7\)
\(\dfrac{3c-7d}{d}=\dfrac{3dk-7d}{d}=\dfrac{d\left(3k-7\right)}{d}=3k-7\)
Do đó: \(\dfrac{3a-7b}{b}=\dfrac{3c-7d}{d}\)
\(A=\dfrac{4a+2b-6b-8a+4b}{\left(2a-b\right)\left(2a+b\right)}:\dfrac{4a^2-b^2+4a^2+b^2}{\left(2a-b\right)\left(2a+b\right)}\)
\(=\dfrac{-4a}{\left(2a-b\right)\left(2a+b\right)}\cdot\dfrac{\left(2a-b\right)\left(2a+b\right)}{8a^2}=\dfrac{-1}{2a}\)