\(\left(x-y\right)^2\ge0\Leftrightarrow x^2+y^2\ge2xy\Leftrightarrow2\left(x^2+y^2\right)\ge\left(x+y\right)^2\)

\(P=4\left(x^2+y^2\right)+\frac{1}{x+y}\ge2\left(x+y\right)^2+\frac{1}{x+y}\)

\(=\frac{31}{16}\left(x+y\right)^2+\frac{1}{16}\left(x+y\right)^2+\frac{1}{2\left(x+y\right)}+\frac{1}{2\left(x+y\right)}\)

\(\ge\frac{31}{16}.2^2+3\sqrt[3]{\frac{1}{16}\left(x+y\right)^2.\frac{1}{2\left(x+y\right)}.\frac{1}{2\left(x+y\right)}}\)

\(=\frac{17}{2}\)

Dấu \(=\)khi \(x=y=1\).

giúp mik

đk : x >= 0; x khác 1

\(a,C=\frac{1}{2\sqrt{x}-2}-\frac{1}{2\sqrt{x}+2}+\frac{\sqrt{x}}{1-x}\)

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

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

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

\(b,x=\frac{4}{9}\left(tm\right)\Rightarrow C=\frac{1}{\frac{4}{9}-1}=-\frac{9}{5}\)

\(c,\left|C\right|=\frac{1}{3}\Rightarrow\left|\frac{1}{x-1}\right|=\frac{1}{3}\)

\(\Rightarrow\orbr{\begin{cases}\frac{1}{x-1}=\frac{1}{3}\\\frac{1}{x-1}=-\frac{1}{3}\end{cases}\Rightarrow\orbr{\begin{cases}x-1=3\\x-1=-3\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=4\left(tm\right)\\x=-2\left(tm\right)\end{cases}}}\)

Bài 5. 

\(P=\left(\frac{\sqrt{x}-2}{x-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\frac{\left(1-x\right)^2}{2}\)(đk: \(x\ge0,x\ne1\)) 

\(=\left[\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(x-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(x-1\right)\left(\sqrt{x}+1\right)}\right].\frac{\left(x-1\right)^2}{2}\)

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

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

\(=-\sqrt{x}\left(\sqrt{x}-1\right)=-x+\sqrt{x}\)

Nếu \(0< x< 1\)thì \(0< \sqrt{x}< 1\Rightarrow x< \sqrt{x}\Leftrightarrow\sqrt{x}-x>0\)

Suy ra \(P>0\).

\(P=-x+\sqrt{x}=-\left(x-\sqrt{x}+\frac{1}{4}\right)+\frac{1}{4}=-\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{1}{4}\le\frac{1}{4}\)

Dấu \(=\)khi \(\sqrt{x}=\frac{1}{2}\Leftrightarrow x=\frac{1}{4}\).

m² -8m -16 =0

m² -2.4m -4\(^2\) =0

(m - 4)\(^2\) = 0

=> m -4 = 0

=> m = 4

HT

m² - 8m - 16 = 0 <=> m² - 8m + 16 - 32 = 0

<=> ( m - 4 )² - ( 4√2 )² = 0 <=> ( m - 4 - 4√2 )( m - 4 + 4√2 ) = 0

<=> m = 4 ± 4√2

\(A=\frac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\frac{2x+\sqrt{x}}{\sqrt{x}}+\frac{2\left(x-1\right)}{\sqrt{x}-1}\)(đk: \(x>0,x\ne1\))

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

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

\(=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2=x-\sqrt{x}+1\)

ĐK : x > 0 , x khác 1

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

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

\(=x-\sqrt{x}+1\)

Trả lời:

a, \(P=\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}-\frac{\sqrt{x}-1}{\sqrt{x}+1}\right):\left(\frac{1}{\sqrt{x}+1}-\frac{\sqrt{x}}{1-\sqrt{x}}+\frac{2}{x-1}\right)\) \(\left(ĐK:x\ge0;x\ne1\right)\)

\(=\left[\frac{\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]:\left(\frac{1}{\sqrt{x}+1}+\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{2}{x-1}\right)\)

\(=\left[\frac{\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right]:\left[\frac{\sqrt{x}-1}{x-1}+\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{x-1}+\frac{2}{x-1}\right]\)

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

\(=\frac{x+2\sqrt{x}+1-x+2\sqrt{x}-1}{x-1}:\frac{\sqrt{x}-1-x-\sqrt{x}+2}{x-1}\)

\(=\frac{4\sqrt{x}}{x-1}:\frac{1-x}{x-1}=\frac{4\sqrt{x}}{x-1}\cdot\frac{x-1}{1-x}=\frac{4\sqrt{x}}{1-x}\)