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Bài 1:
a) \(\sqrt{1-x^2}\)có nghĩa \(\Leftrightarrow\)\(1-x^2\ge0\)
\(\Leftrightarrow\)\(x^2\le1\)
\(\Leftrightarrow\)\(\left|x\right|\le1\)
b) \(\sqrt{\frac{x-2}{x-3}}\)có nghĩa \(\Leftrightarrow\)\(\frac{x-2}{x-3}\ge0\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}x>3\\x\le2\end{cases}}\)
\(M=\frac{\sqrt{x-\sqrt{4\left(x-1\right)}}+\sqrt{x+\sqrt{4\left(x-1\right)}}}{\sqrt{x^2-4\left(x-1\right)}}\cdot\left(1-\frac{1}{x-1}\right)\)
\(M=\frac{\sqrt{x-1-2\sqrt{x-1}+1}+\sqrt{x-1+2\sqrt{x-1}+1}}{\sqrt{x^2-4x+4}}\cdot\frac{x-1-1}{x-1}\)
\(M=\frac{\sqrt{\left(\sqrt{x-1}-1\right)^2}+\sqrt{\left(\sqrt{x-1}+1\right)^2}}{\sqrt{\left(x-2\right)^2}}\cdot\frac{x-2}{x-1}\) (đk: \(x\ge1\)
\(M=\frac{\left|\sqrt{x-1}-1\right|+\left|\sqrt{x-1}+1\right|}{\left|x-2\right|}\cdot\frac{x-2}{x-1}\)
Nếu \(1\le x< 2\) =>\(M=\frac{1-\sqrt{x-1}+\sqrt{x-1}+1}{2-x}\cdot\frac{x-2}{x-1}\)
\(M=-\frac{2}{x-1}\)
Nếu x > 2 => \(M=\frac{\sqrt{x-1}-1+\sqrt{x-1}+1}{x-2}\cdot\frac{x-2}{x-1}\)
\(\frac{2\sqrt{x-1}}{x-1}=\frac{2}{\sqrt{x-1}}\)
\(A=\left(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}}\right)\sqrt{x+\sqrt{x^2-32}}\) với \(x\ge4\sqrt{2}\)
Lời giải:
\(A\sqrt{2}=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})\sqrt{2x+\sqrt{(x-4\sqrt{2})(x+4\sqrt{2})}}\)
\(=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})\sqrt{(\sqrt{x-4\sqrt{2}}+\sqrt{x+4\sqrt{2}})^2}\)
\(=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})(\sqrt{x-4\sqrt{2}}+\sqrt{x+4\sqrt{2}})\)
\(=(\sqrt{x-4\sqrt{2}})^2-(\sqrt{x+4\sqrt{2}})^2=(x-4\sqrt{2})-(x+4\sqrt{2})=-8\sqrt{2}\)
Lời giải:
\(A\sqrt{2}=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})\sqrt{2x+\sqrt{(x-4\sqrt{2})(x+4\sqrt{2})}}\)
\(=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})\sqrt{(\sqrt{x-4\sqrt{2}}+\sqrt{x+4\sqrt{2}})^2}\)
\(=(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}})(\sqrt{x-4\sqrt{2}}+\sqrt{x+4\sqrt{2}})\)
\(=(\sqrt{x-4\sqrt{2}})^2-(\sqrt{x+4\sqrt{2}})^2=(x-4\sqrt{2})-(x+4\sqrt{2})=-8\sqrt{2}\)
Đặt: \(a=\sqrt{2+x};b=\sqrt{2-x}\left(a,b\ge0\right)\)
\(\Rightarrow\hept{\begin{cases}a^2+b^2=4\\a^2-b^2=2x\end{cases}}\)
\(\Rightarrow A=\frac{\sqrt{2+ab}\left(a^3-b^3\right)}{4+ab}=\frac{\sqrt{2+ab}\left(a-b\right)\left(a^2+b^2+ab\right)}{4+ab}\)
\(\Rightarrow A=\frac{\sqrt{2+ab}\left(a-b\right)\left(4+ab\right)}{4+ab}=\sqrt{2+ab}\left(a-b\right)\)
\(\Rightarrow A\sqrt{2}=\sqrt{4+2ab}\left(a-b\right)\)
\(\Rightarrow A\sqrt{2}=\sqrt{\left(a^2+b^2+2ab\right)}\left(a-b\right)=\left(a+b\right)\left(a-b\right)\)
\(\Rightarrow A\sqrt{2}=a^2-b^2=2x\)
\(\Rightarrow A=x\sqrt{2}\)
\(\sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}\)
\(=\sqrt{\left(x+4\right)+4\sqrt{x-4}+4}+\sqrt{\left(x-4\right)+4\sqrt{x-4}+4}\)
\(=\sqrt{\left(\sqrt{x-4}+2\right)^2}+\sqrt{\left(\sqrt{x-4}-2\right)^2}\)
\(=\sqrt{x-4}+2+\left|\sqrt{x-4}-2\right|\) (*) vì x>=4
Với\(\sqrt{x-4}-2\ge0\Rightarrow\sqrt{x-4}\ge2\Leftrightarrow x\ge8\)
Khi đó (*) \(=\sqrt{x-4}+2+\sqrt{x-4}-2=2\sqrt{x-4}\)
Với \(\sqrt{x-4}-2< 0\Rightarrow\sqrt{x-4}< 2\)mà x>=4 \(\Rightarrow4\le x< 8\)
Khi đó (*) \(=\sqrt{x-4}+2-\sqrt{x-4}+2=4\)
Vậy...