Vì a,b,c là số thực dương nên \(\sqrt{a^2}=a;\sqrt{b^2}=b;\sqrt{c^2}\)=c. Vậy ta có

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

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

ta có bdt  \(9\le\left(a+1+b+1+c+1\right)\left(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\right)\)(dễ dàng chứng mình bằng bdt cosi).

=>\(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\ge\)\(\frac{9}{3+\sqrt{3}}\)=> A\(\le3-\frac{9}{3+\sqrt{3}}=\frac{3\sqrt{3}}{3+\sqrt{3}}=\frac{3}{\sqrt{3}+1}\)

dấu = khi a=b=c=\(\frac{\sqrt{3}}{3}\)

đặt \(\sqrt{3x^2+x+2}=a\)

\(a^2+4x^2+x^2-4x+4\)=4ax <=> \(\left(a^2-4ax+4x^2\right)+\left(x^2-4x+4\right)\)=0 <=>(a-2x)²+(x-2)²=0 

=>a=2x và x=2 đồng thởi xảy ra (1)

với x=2 =>a=\(\sqrt{3.4+2+2}\)=4=2x

vậy x=2 thỏa mãn điều kiện (1) =>pt co nghiệm duy nhất x=2

\(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)

\(=\frac{a^2}{b}+b+\frac{b^2}{c}+c+\frac{c^2}{a}+a-a-b-c\)

\(\ge2\sqrt{\frac{a^2b}{b}}+2\sqrt{\frac{b^2c}{c}}+2\sqrt{\frac{c^2a}{a}}-a-b-c\)

\(=2a+2b+2c-a-b-c=a+b+c\)

Dấu '=' xảy ra khi a=b=c

Áp dụng BĐT Cauchy-Schwarz dạng Engle ta có:

\(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\frac{\left(a+b+c\right)^2}{a+b+c}=a+b+c\left(đpcm\right)\)

where is câu hỏi

rút gọn hay tính hay...

\(64-x^2-y^2+xy=64-\left(x^2-xy+y^2\right)\)

                                           = \(8^2-\left(x-y\right)^2\)

                                           =(8-x+y)(8+x-y)

chép mạng

Toán?

a) \(x^2+7x+7y-y^2\)

\(=\left(x+y\right)\left(x-y\right)+7\left(x+y\right)\)

\(=\left(x+y\right)\left(x-y+7\right)\)

b) \(x^2-xy-6y^2\)

\(=-6y^2-3xy+2xy+x^2\)

\(=-3y\left(2y+x\right)+x\left(2y+x\right)\)

\(=\left(x-3y\right)\left(2y+x\right)\)

c) \(x^2-3x^2-6x+8\)

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

\(=\left(x+2\right)\left(x^2-5x+4\right)\)

\(=\left(x+2\right)\left(x^2-4x-x+4\right)\)

\(=\left(x+2\right)\left[x\left(x-4\right)-\left(x-4\right)\right]\)

\(=\left(x+2\right)\left(x-1\right)\left(x-4\right)\)

a)x²+7x+7y-y²=(x-y)(x+y)+7.(x+y)

                       =(x+y)(x-y+7)

b)x²-xy-6y²=x²-xy-4y²-2y²

                   =(x-2y)(x+2y)-y(x-2y)

                   =(x-2y)(x+2y-y)