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a) \(\sqrt{17}-4\) b) \(\sqrt{3}\) c) \(\frac{\sqrt{2}}{2}\) d)\(\frac{\sqrt{x}+1}{\sqrt{x}-1}\) e) \(x-\sqrt{5}\)
f) \(4+2\sqrt{3}\) g) \(3+2\sqrt{2}\) h) \(x+\sqrt{x}+1\) i) \(\frac{3\sqrt{5}-\sqrt{15}}{10}\)
k) \(\sqrt{5}+\sqrt{6}\) i) 5 h) 0 l) \(\sqrt{5}+\sqrt{3}\) m) \(\frac{20\sqrt{3}}{3}\) d) 0
a/ ĐKXĐ: \(x\ge2\)
\(\Leftrightarrow2\sqrt{\left(x-2\right)\left(x+2\right)}-6\sqrt{x-2}+\sqrt{x+2}-3=0\)
\(\Leftrightarrow2\sqrt{x-2}\left(\sqrt{x+2}-3\right)+\sqrt{x+2}-3=0\)
\(\Leftrightarrow\left(2\sqrt{x-2}+1\right)\left(\sqrt{x+2}-3\right)=0\)
\(\Leftrightarrow\sqrt{x+2}-3=0\Rightarrow x=11\)
b/ ĐKXĐ: ....
Đặt \(\left\{{}\begin{matrix}\sqrt{x-2016}=a>0\\\sqrt{y-2017}=b>0\\\sqrt{z-2018}=a>0\end{matrix}\right.\)
\(\frac{a-1}{a^2}+\frac{b-1}{b^2}+\frac{c-1}{c^2}=\frac{3}{4}\)
\(\Leftrightarrow\frac{1}{4}-\frac{a-1}{a^2}+\frac{1}{4}-\frac{b-1}{b^2}+\frac{1}{4}-\frac{c-1}{c^2}=0\)
\(\Leftrightarrow\frac{\left(a-2\right)^2}{a^2}+\frac{\left(b-2\right)^2}{b^2}+\frac{\left(c-2\right)^2}{c^2}=0\)
\(\Leftrightarrow a=b=c=2\Rightarrow\left\{{}\begin{matrix}x=2020\\y=2021\\z=2022\end{matrix}\right.\)
a/ ĐK: \(x\ge0\)
\(\Leftrightarrow\sqrt{3+x}=x^2-3\)
Đặt \(\sqrt{3+x}=a>0\Rightarrow3=a^2-x\) pt trở thành:
\(a=x^2-\left(a^2-x\right)\)
\(\Leftrightarrow x^2-a^2+x-a=0\)
\(\Leftrightarrow\left(x-a\right)\left(x+a+1\right)=0\)
\(\Leftrightarrow x=a\) (do \(x\ge0;a>0\))
\(\Leftrightarrow\sqrt{3+x}=x\Leftrightarrow x^2-x-3=0\)
d/ ĐKXĐ: ...
\(\sqrt{6x^2+1}=\sqrt{2x-3}+x^2\)
\(\Leftrightarrow\sqrt{2x-3}-1+x^2+1-\sqrt{6x^2+1}\)
\(\Leftrightarrow\frac{2\left(x-2\right)}{\sqrt{2x-3}+1}+\frac{x^4+2x^2+1-6x^2-1}{\left(x^2+1\right)^2+\sqrt{6x^2+1}}=0\)
\(\Leftrightarrow\frac{2\left(x-2\right)}{\sqrt{2x-3}+1}+\frac{x^2\left(x+2\right)\left(x-2\right)}{\left(x^2+1\right)^2+\sqrt{6x^2+1}}=0\)
\(\Leftrightarrow\left(x-2\right)\left(\frac{2}{\sqrt{2x-3}+1}+\frac{x^2\left(x+2\right)}{\left(x^2+1\right)^2+\sqrt{6x^2+1}}\right)=0\)
\(\Leftrightarrow x=2\) (phần trong ngoặc luôn dương với mọi \(x\ge\frac{3}{2}\))
ĐK:...
\(g\left(x\right)=\text{}\)\(\frac{3+\sqrt{5}}{\sqrt{3}+\sqrt{3+\sqrt{5}}}+\frac{3-\sqrt{5}}{\sqrt{3}-\sqrt{3-\sqrt{5}}}\)
\(=\frac{\left(3+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{3+\sqrt{5}}\right)}{\left(\sqrt{3}+\sqrt{3+\sqrt{5}}\right)\left(\sqrt{3}-\sqrt{3+\sqrt{5}}\right)}+\frac{\left(3-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{3-\sqrt{5}}\right)}{\left(\sqrt{3}-\sqrt{3-\sqrt{5}}\right)\left(\sqrt{3}+\sqrt{3-\sqrt{5}}\right)}\)
\(=\frac{3\sqrt{3}+\sqrt{15}-3\sqrt{3+\sqrt{5}}-\sqrt{5}\sqrt{3+\sqrt{5}}}{-\sqrt{5}}+\frac{3\sqrt{3}-\sqrt{15}+3\sqrt{3-\sqrt{5}}-\sqrt{5}\sqrt{3-\sqrt{5}}}{\sqrt{5}}\)\(=\frac{-2\sqrt{15}+3\sqrt{3+\sqrt{5}}+3\sqrt{3-\sqrt{5}}+\sqrt{5}\sqrt{3+\sqrt{5}}-\sqrt{5}\sqrt{3-\sqrt{5}}}{\sqrt{5}}\)
\(=\frac{-4\sqrt{15}+3\sqrt{12+4\sqrt{5}}+3\sqrt{12-4\sqrt{5}}+\sqrt{5}\sqrt{12+4\sqrt{5}}-\sqrt{5}\sqrt{12-4\sqrt{5}}}{2\sqrt{5}}\)
\(=\frac{-4\sqrt{15}+3\left(\sqrt{2}+\sqrt{10}\right)+3\left(\sqrt{10}-\sqrt{2}\right)+\sqrt{5}\left(\sqrt{2}+\sqrt{10}\right)-\sqrt{5}\left(\sqrt{10}-\sqrt{2}\right)}{2\sqrt{5}}\)
\(=\frac{-4\sqrt{15}+6\sqrt{10}+2\sqrt{10}}{2\sqrt{5}}=-2\sqrt{3}+4\sqrt{2}\)
