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21 tháng 12 2020
Mk sai từ dòng 3 nhá --
\(=\left(x^2-1\right)\left(\frac{2-\left(x^2-1\right)}{\left(x-1\right)\left(x+1\right)}\right)\)
\(=\frac{\left(x^2-1\right)\left(2-\left(x^2-1\right)\right)}{\left(x-1\right)\left(x+1\right)}=2-x^2+1=3-x^2\)
21 tháng 12 2020
\(\left(x^2-1\right)\left(\frac{1}{x-1}-\frac{1}{x+1}-1\right)\)
\(=\left(x^2-1\right)\left(\frac{x+1}{\left(x-1\right)\left(x+1\right)}-\frac{x-1}{\left(x+1\right)\left(x-1\right)}-\frac{\left(x+1\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\right)\)
\(=\left(x^2-1\right)\left(\frac{-\left(x^2-1\right)}{\left(x-1\right)\left(x+1\right)}\right)\)
\(=\frac{-\left(x-1\right)^2\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}=-\left(x-1\right)\left(x+1\right)=-x^2+1\)
TT
3
5 tháng 1 2023
\(\dfrac{x^2+3\text{x}y+2y^2}{\left(x^3+2\text{x}^2y\right)-\left(xy^2+2y^3\right)}\)
=\(\dfrac{x^2+3\text{x}y+2y^2}{x^2\left(x+2y\right)-y^2\left(x+2y\right)}\)=\(\dfrac{x^2+3\text{x}y+2y^2}{\left(x+2y\right)\left(x^2-y^2\right)}\)
câu 17
14 tháng 2 2022
a/(b+c) + b/(a+c) + c/(a+b) = a^2/(ab+ac) + b^2/(ba+bc) + c^2/(ac+bc) >=
(a+b+c)^2/(2.(ab+bc+ac) (buhihacopxki dạng phân thức)
>= (3.(ab+bc+ac)/(2(ab+bc+ac) =3/2
a^2/(b^2+c^2) + b^2/(a^2+c^2) + c^2/(a^2+b^2) >= (a+b+c)^2/(2.(a^2+b^2+c^2) (buhihacopxki dạng phân thức)
>= 3(a^2+b^2+c^2) / 2(a^2+b^2+c^2) >=3/2
15 tháng 2 2022
\(\Leftrightarrow\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}-\dfrac{3}{2}\ge0\)
\(\Leftrightarrow\left(\dfrac{a}{b+c}-\dfrac{1}{2}\right)+\left(\dfrac{b}{c+a}-\dfrac{1}{2}\right)+\left(\dfrac{c}{a+b}-\dfrac{1}{2}\right)\ge0\)
\(\Leftrightarrow\left(\dfrac{2a-b-c}{2\left(b+c\right)}\right)+\left(\dfrac{2b-a-c}{2\left(a+c\right)}\right)+\left(\dfrac{2c-a-b}{2\left(a+b\right)}\right)\ge0\)
\(\Leftrightarrow\dfrac{a-b+a-c}{2\left(b+c\right)}+\dfrac{b-a+b-c}{2\left(a+c\right)}+\dfrac{c-a+c-b}{2\left(a+b\right)}\ge0\)
\(\Leftrightarrow\dfrac{a-b}{2\left(b+c\right)}+\dfrac{a-c}{2\left(b+c\right)}+\dfrac{b-a}{2\left(a+c\right)}+\dfrac{b-c}{2\left(a+c\right)}+\dfrac{c-a}{2\left(a+b\right)}+\dfrac{c-b}{2\left(a+b\right)}\ge0\)\(\Leftrightarrow\left(a-b\right)\left[\dfrac{1}{2\left(b+c\right)}-\dfrac{1}{2\left(a+c\right)}\right]+\left(a-c\right)\left[\dfrac{1}{2\left(b+c\right)}-\dfrac{1}{2\left(a+b\right)}\right]+\left(b-c\right)\left[\dfrac{1}{2\left(a+c\right)}-\dfrac{1}{2\left(a+b\right)}\right]\ge0\)
ta có: a,b,c là 3 số dương bất kì nên ta giả sử \(a\ge b\ge c\)
\(\Rightarrow a+c\ge b+c\)
\(\Leftrightarrow2\left(a+c\right)\ge2\left(b+c\right)\)
\(\Leftrightarrow\dfrac{1}{2\left(a+c\right)}\le\dfrac{1}{2\left(b+c\right)}\)
\(\Leftrightarrow\dfrac{1}{2\left(a+c\right)}-\dfrac{1}{2\left(b+c\right)}\ge0\)
Mà \(a\ge b\Rightarrow a-b\ge0\)
\(\Rightarrow\left(a-b\right)\left[\dfrac{1}{2\left(b+c\right)}-\dfrac{1}{2\left(a+c\right)}\right]\ge0\left(1\right)\)
Chứng minh tương tự, ta có:
\(\left(a-c\right)\left[\dfrac{1}{2\left(b+c\right)}-\dfrac{1}{2\left(a+b\right)}\right]\ge0\left(2\right)\)
\(\left(b-c\right)\left[\dfrac{1}{2\left(a+c\right)}-\dfrac{1}{2\left(a+b\right)}\right]\ge0\left(3\right)\)
Cộng từng vế (1);(2);(3) \(\Rightarrow\) luôn đúng
\(\Rightarrow\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{3}{2}\)
3 tháng 10 2021
1) \(=\left(9x^2-25y^2\right)-\left(6x-10y\right)=\left(3x-5y\right)\left(3x+5y\right)-2\left(3x-5y\right)=\left(3x-5y\right)\left(3x+5y-2\right)\)
2) \(=9x^2y^2-\left(x^2-2xy+y^2\right)=9x^2y^2-\left(x-y\right)^2=\left(3xy-x+y\right)\left(3xy+x-y\right)\)
DH
1
3 tháng 1 2016
3 - 2x = 3(x+1)-x-2
3-2x-3(x+1)+x+2=0
3-2x-3x-3+x+2=0
-4x+2=0
-4x=-2
x=\(\frac{-2}{-4}\)
x=\(\frac{1}{2}\)
Bài III:
2) (Xem trong ảnh)
Bài 3:
\(\frac{x-5}{x-2}-\frac{x+4}{2x-x^2}\left(ĐK:x\ne2;x\ne0\right)\)
\(=\frac{x-5}{x-2}+\frac{x+4}{x^2-2x}\)
\(=\frac{x.(x-5)}{x.\left(x-2\right)}+\frac{x+4}{x.\left(x-2\right)}\)
\(=\frac{x^2-5x+x+4}{x.\left(x-2\right)}\)
\(=\frac{x^2-4x+4}{x.\left(x-2\right)}\)
\(=\frac{\left(x-2\right)^2}{x.\left(x-2\right)}\)
\(=\frac{x-2}{x}\)
\(\frac{x-3}{x+2}+\frac{4x}{x-3}-\frac{8x+4x^2}{x^2-x-6}\)
\(=\frac{\left(x-3\right)^2+4x.\left(x+2\right)-8x-4x^2}{\left(x-3\right).\left(x+2\right)}\)
\(=\frac{x^2-6x+9+4x^2+8x-8x-4x^2}{\left(x-3\right).\left(x+2\right)}\)
\(=\frac{\left(x-3\right)^2}{\left(x+2\right).\left(x-3\right)}\)
\(=\frac{x-3}{x+2}\)