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

\(\Leftrightarrow\frac{1}{x-3}-\sqrt{x-3}=0\)

Em không chắc đâu ạ!

ĐKXĐ: \(\hept{\begin{cases}x\ne3\\x\ge3\end{cases}}\Rightarrow x>3\)

\(y=\frac{1}{\sqrt{x^2-6x+9}}-\sqrt{x-3}=\frac{1}{\sqrt{\left(x-3\right)^2}}-\sqrt{x-3}\)

Do x > 3 nên x - 3 > 0. Do vậy:\(y=\frac{1}{x-3}-\sqrt{x-3}\)

Đặt \(\sqrt{x-3}=t>0\Rightarrow y=\frac{1}{t^2}-t\)

Theo đề bài suy ra \(y=\frac{1}{t^2}-t=0\Leftrightarrow t=\frac{1}{t^2}\Leftrightarrow t^3=1\Leftrightarrow\orbr{\begin{cases}t=1\\t=-1\left(L\right)\end{cases}}\)

t = 1 \(\Rightarrow\sqrt{x-3}=1\Rightarrow x=4\) (TMĐKXĐ)

Bài 1: 

\(\Leftrightarrow\left(x^2-6x-7\right)^2-\left(3x^2-12x-9\right)^2=0\)

\(\Leftrightarrow\left(3x^2-12x-9-x^2+6x+7\right)\left(3x^2-12x-9+x^2-6x-7\right)=0\)

\(\Leftrightarrow\left(2x^2-6x-2\right)\left(4x^2-18x-16\right)=0\)

\(\Leftrightarrow\left(x^2-3x-1\right)\left(2x^2-9x-8\right)=0\)

hay \(x\in\left\{\dfrac{3+\sqrt{13}}{2};\dfrac{3-\sqrt{13}}{2};\dfrac{9+\sqrt{145}}{4};\dfrac{9-\sqrt{145}}{4}\right\}\)

SEIFWJNHGRHFQ24FTW

a) ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\y\ge0\\x\ne y\end{matrix}\right.\)

Gọi biểu thức trên là A , ta có:

\(A=\frac{2\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}+\frac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}-\frac{3\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\\ =\frac{2\sqrt{x}-2\sqrt{y}+\sqrt{x}+\sqrt{y}-3\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\\ =\frac{-\sqrt{y}}{x-y}\left(=\frac{\sqrt{y}}{y-x}\right)\)

b) Với x=4 ; y=9 ta có:

\(A=\frac{\sqrt{9}}{9-4}=\frac{3}{5}\)

c) Ta có: với x>y>0 thì A<=>\(\left\{{}\begin{matrix}\sqrt{y}>0\\x>y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\sqrt{y}>0\\y-x< 0\end{matrix}\right.\Leftrightarrow A< 0\)

Vậy A<0 với mọi x>y>0

a) ĐK: \(x>2009;y>2010;z>2011\)

\(\Leftrightarrow\frac{\sqrt{x-2009}-1}{x-2009}-\frac{1}{4}+\frac{\sqrt{y-2010}-1}{y-2010}-\frac{1}{4}+\frac{\sqrt{z-2011}-1}{z-2011}-\frac{1}{4}=0\)

\(\Leftrightarrow\frac{-\left(\sqrt{x-2009}-2\right)^2}{4\left(x-2009\right)}+\frac{-\left(\sqrt{y-2010}-2\right)^2}{4\left(y-2010\right)}+\frac{-\left(\sqrt{z-2011}-2\right)^2}{4\left(z-2011\right)}=0\left(1\right)\)

Dễ thấy với đkxđ thì \(VT\left(1\right)\le0\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}\sqrt{x-2009}=2\\\sqrt{y-2010}=2\\\sqrt{z-2011}=2\end{cases}\Leftrightarrow\hept{\begin{cases}x=2013\\y=2014\\z=2015\end{cases}\left(tm\right)}}\)

\(\sqrt{x^2-9}+\sqrt{x^2-6x+9}=0\)(*)

\(ĐK:\orbr{\begin{cases}x\ge3\\x\le-3\end{cases}}\)

(*)\(\Leftrightarrow\sqrt{\left(x+3\right)\left(x-3\right)}+\sqrt{\left(x-3\right)^2}=0\)

\(\Leftrightarrow\sqrt{x-3}\left(\sqrt{x+3}+\sqrt{x-3}\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=3\left(tm\right)\\\sqrt{x+3}+\sqrt{x-3}=0\end{cases}}\)

Xét phương trình\(\sqrt{x+3}+\sqrt{x-3}=0\)(**) có \(\sqrt{x+3}\ge0;\sqrt{x-3}\ge0\)nên (**) xảy ra khi \(\hept{\begin{cases}\sqrt{x+3}=0\\\sqrt{x-3}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-3\\x=3\end{cases}}\left(L\right)\)

Vậy phương trình có một nghiệm duy nhất là 3

a. ĐK : x > 2009 ; y > 2010 ; z > 2011 

Pt <=> \(\frac{1-\sqrt{x-2009}}{x-2009}+\frac{1-\sqrt{y-2010}}{y-2010}+\frac{1-\sqrt{z-2011}}{z-2011}=-\frac{3}{4}\)

\(\Leftrightarrow\left(\frac{1}{x-2009}-\frac{1}{\sqrt{x-2009}}+\frac{1}{4}\right)+\left(\frac{1}{y-2010}-\frac{1}{\sqrt{y-2010}}+\frac{1}{4}\right)\)

\(\left(\frac{1}{z-2011}-\frac{1}{\sqrt{z-2011}}+\frac{1}{4}\right)=0\)

\(\Leftrightarrow\left(\frac{1}{\sqrt{x-2009}}-\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{y-2010}}-\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{z-2011}}-\frac{1}{2}\right)^2=0\)

\(\Leftrightarrow\hept{\begin{cases}\left(\frac{1}{\sqrt{x-2009}}-\frac{1}{2}\right)^2=0\\\left(\frac{1}{\sqrt{y-2010}}-\frac{1}{2}\right)^2=0\\\left(\frac{1}{\sqrt{z-2011}}-\frac{1}{2}\right)^2=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\frac{1}{\sqrt{x-2009}}=\frac{1}{2}\\\frac{1}{\sqrt{y-2010}}=\frac{1}{2}\\\frac{1}{\sqrt{z-2011}}=\frac{1}{2}\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}\sqrt{x-2009}=2\\\sqrt{y-2010}=2\\\sqrt{z-2011}=2\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=2013\\y=2014\\z=2015\end{cases}}\)( tmđk )

b. ĐK : x² - 9 \(\ge\)0 <=> x²\(\ge\)9 <=> - 3\(\le\)x\(\le\)3

\(\sqrt{x^2-9}+\sqrt{x^2-6x+9}=0\)

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

\(\Leftrightarrow\sqrt{x-3}\left(\sqrt{x+3}+\sqrt{x-3}\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x-3}=0\\\sqrt{x+3}+\sqrt{x-3}=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=3\left(tmdk\right)\\\sqrt{x+3}+\sqrt{x-3}=0\end{cases}}\)

TH :\(\sqrt{x+3}+\sqrt{x-3}=0\)

Vì \(\sqrt{x+3}+\sqrt{x-3}\ge0\forall x\). Dấu "=" xảy ra <=> \(\Leftrightarrow\orbr{\begin{cases}\sqrt{x+3}=0\\\sqrt{x-3}=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=-3\\x=3\end{cases}}\)( mâu thuẫn )

Vậy pt có nghiệm duy nhất là x = 3

2/ x² - 6x + 4 + \(2\sqrt{2x-1}\)= 0

<=> (x² - 4x + 4) - (2x - 1 - \(2\sqrt{2x-1}\)+1) = 0

<=> (x - 2)² - (1 - \(\sqrt{2x-1}\))² = 0

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

Làm tiếp nhé

câu mik muốn hỏi là câu 1 bn giúp mik

Câu 1/ 

\(\frac{x\sqrt{y}+y\sqrt{x}}{x+y}-\frac{x+y}{2}\le\frac{x\sqrt{y}+y\sqrt{x}}{2\sqrt{xy}}-\frac{x+y}{2}\)

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

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

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

\(\le\frac{\sqrt{x}+\sqrt{y}}{2}-\frac{\sqrt{x}+\sqrt{y}}{2}+\frac{1}{4}=\frac{1}{4}\)