a/ rút gọn A

b/tìm a để giá trị của A đạt GTLN

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

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

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

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

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

mình nghĩ bài này sai đề, 

ĐÚng phải là\(\sqrt[3]{2+\sqrt{3}}\)

(   KHÔNG CHẮC NỮA   :D   )

\(\text{sai đề chú ơi}\)

\(A=\left(1+\frac{5}{\sqrt{x}-2}\right).\left(\sqrt{x}-\frac{x+2\sqrt{x}+4}{\sqrt{x}+3}\right).\)

\(=\frac{\sqrt{x}-2+5}{\sqrt{x}-2}.\frac{x+3\sqrt{x}-x-2\sqrt{x}-4}{\sqrt{x}+3}\)

\(=\frac{\sqrt{x}+3}{\sqrt{x}-2}.\frac{\sqrt{x}-4}{\sqrt{x}+3}=\frac{\sqrt{x}-4}{\sqrt{x}-2}\)

ĐK \(x\ge-2\)

pT<=> \(2\left(x+1\right)\sqrt{x+2}+2\left(x+6\right)\sqrt{x+7}=2x^2+14x+24\)

<=>\(\left(x+1\right)\left(x+2-2\sqrt{x+2}\right)+\left(x+6\right)\left(x+4-2\sqrt{x+7}\right)+x-2=0\)

<=>\(\frac{\left(x+1\right)\left(x^2-4\right)}{x+2+2\sqrt{x+2}}+\frac{\left(x+6\right)\left(x^2+4x-12\right)}{x+4+2\sqrt{x+7}}+x-2=0\forall x>-2\)

=> \(\orbr{\begin{cases}x=2\\\frac{\left(x+1\right)\left(x+2\right)}{x+2+2\sqrt{x+2}}\end{cases}}+\frac{x+6}{x+4+2\sqrt{x+7}}+1=0\left(2\right)\)

Pt (2) + \(x\ge-1\)=> \(VT>0\)=> PT (2) vô nghiệm

+  \(-2< x\le-1\)=> \(\frac{\left(x+1\right)\left(x+2\right)}{x+2+2\sqrt{x+2}}>-1\)=> \(VT>0\)=> PT vô nghiệm

Vậy x=2

a, c.Câu hỏi của Nữ hoàng sến súa là ta - Toán lớp 9 - Học toán với OnlineMath

a)+) \(A=\sqrt{2x^2-3x+1}=\sqrt{2x^2-2x-x+1}\)

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

Để A có nghĩa thì \(\hept{\begin{cases}2x-1\ge0\\x-1\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge\frac{1}{2}\\x\ge1\end{cases}}\Leftrightarrow x\ge1\)

hoặc \(\hept{\begin{cases}2x-1\le0\\x-1\le0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\le\frac{1}{2}\\x\le1\end{cases}}\Leftrightarrow x\le\frac{1}{2}\)

A có nghĩa\(\Leftrightarrow\orbr{\begin{cases}x\ge1\\x\le\frac{1}{2}\end{cases}}\)

+) B có nghĩa\(\Leftrightarrow\hept{\begin{cases}x-1\ge0\\2x-1\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge1\\x\ge\frac{1}{2}\end{cases}}\Leftrightarrow x\ge1\)

c) \(A=B\Leftrightarrow\sqrt{\left(x-1\right)\left(2x-1\right)}=\sqrt{x-1}.\sqrt{2x-1}\)

\(\Leftrightarrow\hept{\begin{cases}x-1\ge0\\2x-1\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge1\\x\ge\frac{1}{2}\end{cases}}\Leftrightarrow x\ge1\)

Vậy \(x\ge1\)thì A = B

d) \(x\le\frac{1}{2}\)