Có \(A=\frac{2x+1}{x^2+3}\)

\(\Leftrightarrow Ax^2+3A=2x+1\)

\(\Leftrightarrow Ax^2-2x+3A-1=0\)

Có \(\Delta'=1-A\left(3A-1\right)\)

         \(=1-3A^2+A\)

Pt có nghiệm khi \(\Delta'\ge0\Leftrightarrow-3A^2+A+1\ge0\)

                                        \(\Leftrightarrow\frac{1-\sqrt{13}}{6}\le A\le\frac{1+\sqrt{13}}{6}\)

Nên \(A_{min}=\frac{1-\sqrt{13}}{6}\)

Dấu "=" \(\Leftrightarrow\frac{2x+1}{x^2+3}=\frac{1-\sqrt{13}}{6}\)

Giải ra tìm đc x

Vậy .............

=(a+b-c)(b+c-a)(a+c-b)

M xác định

\(\Leftrightarrow\hept{\begin{cases}x-1\ne0\\x^2-x\ne0\end{cases}\Leftrightarrow}\hept{\begin{cases}x\ne1\\x\left(x-1\right)\ne0\end{cases}\Leftrightarrow\hept{\begin{cases}x\ne1\\x\ne0;x\ne1\end{cases}}\Leftrightarrow}\hept{\begin{cases}x\ne1\\x\ne0\end{cases}}\)

Vậy ĐKXĐ của M là \(\hept{\begin{cases}x\ne1\\x\ne0\end{cases}}\)

\(M=\frac{3}{x-1}+\frac{1}{x^2-x}=\frac{3}{x-1}+\frac{1}{x\left(x-1\right)}=\frac{3x}{x\left(x-1\right)}+\frac{1}{x\left(x-1\right)}=\frac{3x+1}{x\left(x-1\right)}\)

Thay x=5 ta có: 

\(M=\frac{3.5+1}{5\left(5-1\right)}=\frac{15+1}{5.4}=\frac{16}{20}=\frac{4}{5}\)

Vậy \(M=5\)tại  x=5

\(M=0\)

\(\Leftrightarrow\frac{3x+1}{x\left(x-1\right)}=0\Leftrightarrow3x+1=0\Leftrightarrow x=-\frac{1}{3}\)( thỏa mãn đkxđ)

Vậy với \(x=-\frac{1}{3}\)thì \(M=0\)

\(M=-1\)

\(\Leftrightarrow\frac{3x+1}{x\left(x-1\right)}=-1\Leftrightarrow3x+1=-x^2+x\Leftrightarrow x^2+2x+1=0\Leftrightarrow\left(x+1\right)^2=0\Leftrightarrow x=-1\)

Vậy với \(x=-1\)thì \(M=-1\)

 Bài làm :

 (2n+5)²-25  = (2n+5)²-25

                    = (2n+5) . (2n+5) - 25

                    = (2n.2n+2n. 5) + (5.2n + 5.5)-25

                    = (2n²+ 10n) + (10n+25)-25

                     = 2n² + 10n '+ 10n + 25 - 25

                     = 2n²  + (10n+10n) +0

                     = 2n²   + 10n .2 

                     = 2n²    + 20n 

                     =( 2².n²) +( 2².5.n)

                     = 4.n.n + 4.5.n

                     = 4.n.n + 4 .(4+1) .n

                     = 4.n.n + (4.4 + 4).n

                     = 4.n.n + 4.4.n + 4.n

                     = (4.n.n +4.n.1) + 4.4.n

                     = 4n.(n+1) + 4².n

                     = 4n.(n+1) + 8.2.n

                     = 4n.2.(n+1)+8n

                     =  8n. (n+1) +8n                   

       Vì \(\hept{\begin{cases}8n.\left(n+1\right)⋮8\\8n⋮8\end{cases}}\)             => 8n.(n+1)+8n\(⋮\)8 => (2n+5)²-25\(⋮\)8

Vậy (2n+5)²-25\(⋮\)8

BĐT Nesbitt  nhé ko phải Nesbit đâu .V
Bđt đấy đây: Cho a,b,c dương

CMR: \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\ge\frac{3}{2}\)

Giải

Ta có: \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=\left(\frac{a}{b+c}+1\right)+\left(\frac{b}{a+c}+1\right)+\left(\frac{c}{a+b}+1\right)-3\)

      \(=\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\)

       \(=\left(a+b+c\right)\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)-3\)

       \(=\frac{1}{2}.\left[\left(a+b\right)+\left(b+c\right)+\left(c+a\right)\right]\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)-3\)

Áp dụng bđt Cô-si cho 3 số dương được

\(\frac{1}{2}\left[\left(a+b\right)+\left(b+c\right)+\left(c+a\right)\right]\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)-3\)

            \(\ge\frac{1}{2}.3\sqrt[3]{\left(a+b\right)\left(b+c\right)\left(c+a\right)}.3.\sqrt[3]{\frac{1}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}}-3\)

                 \(=\frac{1}{2}.9.\sqrt[3]{\frac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}}-3\)

                  \(=\frac{9}{2}-3\)

                   \(=\frac{3}{2}\)

Dấu "='' xảy ra <=> a=b=c

Vậy ...........

BĐT Nesbit: Với a,b,c dương:

\(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\)

\(BĐT\Leftrightarrow\left(\frac{a}{b+c}+1\right)+\left(\frac{b}{c+a}+1\right)+\left(\frac{c}{a+b}+1\right)\ge\frac{9}{2}\)

\(\Leftrightarrow2\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)\ge9\)

\(\Leftrightarrow2\left[\left(a+b\right)+\left(b+c\right)+\left(c+a\right)\right]\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)\ge9\)

Dùng bất đẳng thức cô si hai lần vào vế trái sẽ có điều cần chứng minh.

\(\frac{x^4+x^3-x^2-2x-2}{x^4+2x^3-x^2-4x-2}=\frac{\left(x^4-x^2-2\right)+\left(x^3-2x\right)}{\left(x^4-x^2-2\right)+\left(2x^3-4x\right)}\)

\(=\frac{\left(x^2-2\right)\left(x^2+1\right)+x\left(x^2-2\right)}{\left(x^2-2\right)\left(x^2+1\right)+2x\left(x^2-2\right)}=\frac{\left(x^2-2\right)\left(x^2+x+1\right)}{\left(x^2-2\right)\left(x^2+2x+1\right)}\)

\(=\frac{x^2+x+1}{\left(x+1\right)^2}\)

\(F\left(x\right)=\frac{x^4+x^3-x^2-2x-2}{x^4+2x^3-x^2-4x-2}\)

\(=\frac{\left(x^4+x^3+x^2\right)-2x^2-2x-2}{\left(x^4+2x^3+x^2\right)-\left(2x^2+4x+2\right)}\)

\(=\frac{x^2\left(x^2+x+1\right)-2\left(x^2+x+1\right)}{x^2\left(x^2+2x+1\right)-2\left(x^2+2x+1\right)}=\frac{x^2+x+1}{x^2+2x+1}\)

\(a,10n^2+n-10⋮n-1\)

\(10n^2-10n+11n-11+1⋮n-1\)

Do \(10n\left(n-1\right)⋮n-1;11\left(n-1\right)⋮n-1\Rightarrow1⋮n-1\)

\(\Rightarrow n-1\in\left(1;-1\right)\)

\(\Rightarrow n\in\left(2;0\right)\)

\(b,25n^2-97n+11⋮n-4\)

\(25n^2-100n+3n-12+23⋮n-4\)

\(25n\left(n-4\right)+3\left(n-4\right)+23⋮n-4\)

\(\Rightarrow23⋮n-4\)

\(\Rightarrow n-4\in\left(1;-1;23;-23\right)\)

\(\Rightarrow n\in\left(5;3;27;-19\right)\)