Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
a) đk: \(x\ge0;x\ne\left\{\frac{1}{4};1\right\}\)
\(P=\left(\frac{2x\sqrt{x}+x-\sqrt{x}}{x\sqrt{x}-1}-\frac{x+\sqrt{x}}{x-1}\right)\cdot\frac{x-1}{2x+\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
\(P=\left[\frac{\left(2x+\sqrt{x}-1\right)\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}+1\right)\sqrt{x}}{x-1}\right]\cdot\frac{x-1}{2x+\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
\(P=\frac{\left(x-1\right)\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}+1\right)\sqrt{x}}{2x+\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
\(P=\frac{x+\sqrt{x}}{x+\sqrt{x}+1}-\frac{\sqrt{x}}{2\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
\(P=\frac{x+\sqrt{x}}{x+\sqrt{x}+1}\)
b) Ta có:
\(P=\frac{x+\sqrt{x}}{x+\sqrt{x}+1}=\frac{\left(x+\sqrt{x}+1\right)-1}{x+\sqrt{x}+1}=1-\frac{1}{x+\sqrt{x}+1}\)
Mà \(x+\sqrt{x}\ge0\left(\forall x\right)\)
\(\Leftrightarrow x+\sqrt{x}+1\ge1\left(\forall x\right)\)
\(\Leftrightarrow\frac{1}{x+\sqrt{x}+1}\le1\left(\forall x\right)\)
\(\Leftrightarrow P=1-\frac{1}{x+\sqrt{x}+1}\ge0\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(x+\sqrt{x}=0\Leftrightarrow x=0\)
Vậy Min(P) = 0 khi x = 0
ĐKXĐ: \(x\ge0\) và \(x\ne9\)
a/ \(\frac{x\sqrt{x}-3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}-\frac{2\sqrt{x}-6}{\sqrt{x}+1}-\frac{\sqrt{x}+3}{\sqrt{x}-3}\)
\(=\frac{x\sqrt{x}-3-\left(2\sqrt{x}-6\right)\left(\sqrt{x}-3\right)-\left(\sqrt{x}+3\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{x\sqrt{x}-3-2x+12\sqrt{x}-18-x-4\sqrt{x}-3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{x\sqrt{x}+8\sqrt{x}-3x-24}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}=\frac{\sqrt{x}\left(x+8\right)-3\left(x+8\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{\left(x+8\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}=\frac{x+8}{\sqrt{x}+1}\)
b/ Thay \(x=14-6\sqrt{5}\) vào P ta được:
\(P=\frac{14-6\sqrt{5}+8}{\sqrt{14-6\sqrt{5}}+1}=\frac{22-6\sqrt{5}}{3-\sqrt{5}+1}=\frac{22-6\sqrt{5}}{4-\sqrt{5}}\)
Câu 1: Điều kiện xác định
a/ \(\hept{\begin{cases}x\ge0\\x-9\ne0\end{cases}\Rightarrow\hept{\begin{cases}x\ge0\\x\ne9\end{cases}}}\)
b/ \(Q=\frac{\sqrt{x}-1}{x}+\frac{2\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}+1\right)}\)
\(\hept{\begin{cases}x>0\\\sqrt{x}+1\ne0\end{cases}\Rightarrow x>0}\)
c/ \(\hept{\begin{cases}x\ge0\\x-5\ne0\end{cases}\Rightarrow\hept{\begin{cases}x\ge0\\x\ne5\end{cases}}}\)
Câu 2:
a/ ĐKXĐ: \(\hept{\begin{cases}x>0\\\sqrt{x}-1\ne0\end{cases}\Rightarrow\hept{\begin{cases}x>0\\x\ne1\end{cases}}}\)
b/ \(P=\left(1+\frac{1}{\sqrt{x}-1}\right)-\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}-1\right)+\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=\frac{x-\sqrt{x}+\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}+1}{\sqrt{x}}\)
c/ Thay x = 25 vào P ta được: \(P=\frac{\sqrt{25}+1}{\sqrt{25}}=\frac{6}{5}\)
d/ Ta có: \(P=\frac{\sqrt{5+2\sqrt{6}}+1}{\sqrt{5+2\sqrt{6}}}=\frac{\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}+1}{\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}}=\frac{\sqrt{3}+\sqrt{2}+1}{\sqrt{3}+\sqrt{2}}\)