Bài 3 :
\(\Leftrightarrow\sqrt{9x^2-6x+1}=\sqrt{\left(3x-1\right)^2}=\left|3x-1\right|=5\)
\(\Leftrightarrow\left[{}\begin{matrix}3x-1=5\\3x-1=-5\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-\dfrac{4}{3}\end{matrix}\right.\)
Vậy ...
Bài 5 :
Ta có :\(x-5\sqrt{x}+7=x-2.\sqrt{x}.\dfrac{5}{2}+\dfrac{25}{4}+\dfrac{3}{4}=\left(\sqrt{x}-\dfrac{5}{2}\right)^2+\dfrac{3}{4}\)
Thấy : \(\left(\sqrt{x}-\dfrac{5}{2}\right)^2\ge0\)
\(\Rightarrow\left(\sqrt{x}-\dfrac{5}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)
\(\Rightarrow P=\dfrac{1}{x-5\sqrt{x}+7}=\dfrac{1}{\left(\sqrt{x}-\dfrac{5}{2}\right)^2+\dfrac{3}{4}}\le\dfrac{1}{\dfrac{3}{4}}=\dfrac{4}{3}\)
Vậy \(Max_P=\dfrac{4}{3}\Leftrightarrow\sqrt{x}-\dfrac{5}{2}=0\Leftrightarrow x=\dfrac{25}{4}\)
Bài 1:
a) Ta có: \(\sqrt{25}\cdot\sqrt{144}+\sqrt[3]{-27}-\sqrt[3]{216}\)
\(=5\cdot12-3-6\)
\(=60-9=51\)
b) Ta có: \(\sqrt{8.1\cdot360}\)
\(=\sqrt{8.1\cdot10\cdot36}\)
\(=\sqrt{81\cdot36}\)
\(=9\cdot6=54\)
Bài 2:
a) Ta có: \(\sqrt{80}-\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{3\dfrac{1}{5}}\)
\(=4\sqrt{5}-\sqrt{5}+2+\dfrac{4}{\sqrt{5}}\)
\(=3\sqrt{5}+2+\dfrac{4\sqrt{5}}{5}\)
\(=\dfrac{10+19\sqrt{5}}{5}\)
b) Ta có: \(\dfrac{\sqrt{6}-\sqrt{3}}{1-\sqrt{2}}+\dfrac{3+6\sqrt{3}}{\sqrt{3}}-\dfrac{13}{\sqrt{3}+4}\)
\(=\dfrac{-\sqrt{3}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}+\dfrac{\sqrt{3}\left(\sqrt{3}+6\right)}{\sqrt{3}}-\dfrac{13\left(4-\sqrt{3}\right)}{\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)}\)
\(=-\sqrt{3}+\sqrt{3}+6-4+\sqrt{3}\)
\(=2+\sqrt{3}\)
Bài 3:
Ta có: \(\sqrt{9x^2+6x+1}=5\)
\(\Leftrightarrow\sqrt{\left(3x+1\right)^2}=5\)
\(\Leftrightarrow\left|3x+1\right|=5\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+1=5\\3x+1=-5\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}3x=4\\3x=-6\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{4}{3}\\x=-2\end{matrix}\right.\)
Vậy: \(S=\left\{\dfrac{4}{3};-2\right\}\)
Bài 4:
a) ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)
b) Ta có: \(A=\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{8\sqrt{x}}{x-1}\right):\dfrac{4\sqrt{x}-8}{1-x}\)
\(=\left(\dfrac{\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}-1\right)^2-8\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\dfrac{4\sqrt{x}-8}{-\left(x-1\right)}\)
\(=\dfrac{x+2\sqrt{x}+1-x+2\sqrt{x}-1-8\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\dfrac{-\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{4\left(\sqrt{x}-2\right)}\)
\(=\dfrac{4\sqrt{x}}{4\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}-2}\)
