a) Ta có : 

VT : \(\dfrac{\left(x\sqrt{y}+y\sqrt{x}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}\)

\(=\dfrac{\left(\sqrt{x^2y}+\sqrt{xy^2}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}\)

\(=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)\)

\(=\left(\sqrt{x}\right)^2-\left(\sqrt{y}\right)^2=x-y\)  với \(x>0;y>0\)

VT\(=\)VP nên đẳng thức được chứng minh.

b) Vì \(x>0\) nên \(\sqrt{x^3}=\left(\sqrt{x}\right)^3\)

Ta có : 

VT \(\dfrac{\sqrt{x^3}-1}{\sqrt{x}-1}=\dfrac{\left(\sqrt{x}\right)^3-1^3}{\sqrt{x}-1}=\dfrac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}-1}\)

\(=x+\sqrt{x}+1\) với \(x\ge0;x\ne1\)

VT\(=\)VP nên đẳng thức được chứng minh.

\(\dfrac{1}{x-1}+\dfrac{6}{3x+5}=\dfrac{2}{x+2}+\dfrac{1}{x+3}\) (ĐK: \(x\notin\left\{1,-\dfrac{5}{3},-2,-3\right\}\))

\(\Rightarrow\left(3x+5\right)\left(x+2\right)\left(x+3\right)+6\left(x-1\right)\left(x+2\right)\left(x+3\right)=2\left(x-1\right)\left(3x+5\right)\left(x+3\right)+\left(x-1\right)\left(3x+5\right)\left(x+2\right)\)

\(\Leftrightarrow7x^2+24x+17=0\)

\(\Leftrightarrow\left(7x+17\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-17}{7}\\x=-1\end{matrix}\right.\) (thỏa mãn) 

Bạn nên viết đề bằng công thức toán (biểu tượng $\sum$ góc trái khung soạn thảo) để được hỗ trợ tốt hơn!

\(\Leftrightarrow\left\{{}\begin{matrix}2x^2+4x+y^3+3=0\\x^2y^3+y=2x\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x^2+2\left(x^2y^3+y\right)+y^3+3=0\left(1\right)\\x^2y^3+y=2x\left(2\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow2x^2+2x^2y^3+2y+y^3+3=0\Leftrightarrow\left(y+1\right)\left(2x^2y^2-2x^2y+y^2+2x-y+3\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}y=-1\Rightarrow x=-1\\2x^2y^2-2x^2y+y^2+2x^2-y+3=0\left(3\right)\end{matrix}\right.\)

\(\left(3\right)\Leftrightarrow2x^2\left(y^2-y+2\right)+y^2-y+3=0\)

\(\Rightarrow a=y^2-y+2=\left(y-\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\)

\(\Delta=0-4.2\left(y^2-y+2\right)\left(y^2-y+3\right)=-8\left(y^2-y+2\right)\left(y^2-y+3\right)\)

\(y^2-y+3=\left(y-\dfrac{1}{2}\right)^2+\dfrac{11}{4}>0\)

\(\Rightarrow\Delta=-8\left(y^2-y+2\right)\left(y^2-y+3\right)< 0\)

\(\Rightarrow\left(3\right)\) không tồn tại nghiệm (x;y) nào

do đó hpt có  nghiệm x=y=-1

Lời giải:
ĐKXĐ: $x\neq -1$

PT $\Leftrightarrow (x-\frac{x}{x+1})^2+4=\frac{5x^2}{x+1}$

$\Leftrightarrow (\frac{x^2}{x+1})^2+4=\frac{5x^2}{x+1}$
Đặt $\frac{x^2}{x+1}=a$ thì pt trở thành:
$a^2+4=5a$

$\Leftrightarrow (a-1)(a-4)=0$

$\Leftrightarrow a=1$ hoặc $a=4$

Nếu $a=1\Leftrightarrow \frac{x^2}{x+1}=1$

$\Rightarrow x^2-x-1=0$

$\Leftrightarrow x=\frac{1\pm \sqrt{5}}{2}$

Nếu $a=4\Leftrightarrow \frac{x^2}{x+1}=4$

$\Rightarrow x^2-4x-4=0$

$\Leftrightarrow x=2\pm 2\sqrt{2}$

Lời giải:
ĐKXĐ: $a>0; a\neq 1$

\(A=\left[\frac{\sqrt{a}(\sqrt{a}-1)}{\sqrt{a}-1}-\frac{\sqrt{a}+1}{\sqrt{a}(\sqrt{a}+1)}\right]:\frac{\sqrt{a}+1}{a\sqrt{a}}\)

\(=(\sqrt{a}-\frac{1}{\sqrt{a}}).\frac{a\sqrt{a}}{\sqrt{a}+1}=\frac{a-1}{\sqrt{a}}.\frac{a\sqrt{a}}{\sqrt{a}+1}=\frac{(\sqrt{a}-1)(\sqrt{a}+1)}{\sqrt{a}}.\frac{a\sqrt{a}}{\sqrt{a}+1}=a(\sqrt{a}-1)\)

ta có \(\sqrt{\left(a+c\right)\left(a+b\right)}\ge a+\sqrt{bc}\left(1\right)\)

thật vậy \(\left(1\right)\Leftrightarrow\left(a+c\right)\left(a+b\right)\ge a^2+2a\sqrt{bc}+bc\)

\(\Leftrightarrow ab+ac\ge2a\sqrt{bc}\Leftrightarrow b+c\ge2\sqrt{bc}\)(đúng theo BĐT cosi)

cminh tương tự \(\Rightarrow\sqrt{\left(b+c\right)\left(b+a\right)}\ge b+\sqrt{ac};\sqrt{\left(c+a\right)\left(c+b\right)}\ge c+\sqrt{ab}\)

\(\Rightarrow\dfrac{a}{\sqrt{\left(a+c\right)\left(a+b\right)}}\le\dfrac{a}{a+\sqrt{bc}}=\dfrac{1}{1+\dfrac{\sqrt{bc}}{a}}\)

\(tt\Rightarrow P\le\dfrac{1}{1+\dfrac{\sqrt{bc}}{a}}+\dfrac{1}{1+\dfrac{\sqrt{ac}}{b}}+\dfrac{1}{1+\dfrac{\sqrt{ab}}{c}}\)

\(đặt\left(\dfrac{\sqrt{bc}}{a};\dfrac{\sqrt{ac}}{b};\dfrac{\sqrt{ab}}{c}\right)=\left(x;y;z\right)\Rightarrow xyz=1\)

\(\Rightarrow P\le\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\)

ta đi chứng minh \(\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\le\dfrac{3}{2}\)

\(\Leftrightarrow2\left(y+1\right)\left(z+1\right)+2\left(x+1\right)\left(z+1\right)+2\left(x+1\right)\left(y+1\right)\le3\left(x+1\right)\left(y+1\right)\left(z+1\right)\)

\(\Leftrightarrow2xy+2xz+2yz+4x+4y+4z+6\le3xyz+3+3xy+3xz+3yz+3x+3y+3z\)

ủa đến đây theo cách làm bth đúng rồi mà sao không ra nhỉ bạn xem lại hộ mình giống bài  n ày mình từng làm r

https://hoc24.vn/vip/289470733648/page-12

Toán 9

Ta có: \(\dfrac{2}{x^2+y^2}=\dfrac{x^2+y^2+z^2}{x^2+y^2}=1+\dfrac{z^2}{x^2+y^2}\le1+\dfrac{z^2}{2xy}\)(bđt cosi)

CMTT: \(\dfrac{2}{y^2+z^2}\le1+\dfrac{x^2}{2yz}\); \(\dfrac{2}{z^2+x^2}\le1+\dfrac{y^2}{2xz}\)

=> \(\dfrac{2}{x^2+y^2}+\dfrac{2}{y^2+z^2}+\dfrac{2}{z^2+x^2}\le3+\dfrac{z^2}{2xy}+\dfrac{x^2}{2yz}+\dfrac{y^2}{2xz}=3+\dfrac{x^3+y^3+z^3}{2xyz}\) (Đpcm)

Ta có: $\frac{2}{x^2+y^2}=\frac{x^2+y^2+z^2}{x^2+y^2}=1+\frac{z^2}{x^2+y^2}\le 1+\frac{z^2}{2xy}$(bđt cosi)

CMTT: $\frac{2}{y^2+z^2}\le 1+\frac{x^2}{2yz}$; $\frac{2}{z^2+x^2}\le 1+\frac{y^2}{2xz}$

=> $\frac{2}{x^2+y^2}+\frac{2}{y^2+z^2}+\frac{2}{z^2+x^2}\le 3+\frac{z^2}{2xy}+\frac{x^2}{2yz}+\frac{y^2}{2xz}=3+\frac{x^3+y^3+z^3}{2xyz}$ ( Đpcm )