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TM
31 tháng 7 2017
\(=\left(\frac{\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}\right).\frac{\left(x-1\right)^2}{2}\)
\(=\left(\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\right).\frac{\left(x-1\right)^2}{2}\)
\(=\frac{\left(x-\sqrt{x}-2\right)-\left(x+\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}.\frac{\left(x-1\right)^2}{2}\)
\(=\frac{-2\sqrt{x}}{\left(x-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(x-1\right)^2}{2}\)\(=\frac{-\sqrt{x}}{\sqrt{x}+1}.\left(x-1\right)=\frac{-x\sqrt{x}+\sqrt{x}}{\sqrt{x}+1}\)
\(=\left(\frac{\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}\right).\frac{\left(x-1\right)^2}{2}\)
\(=\left(\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\right).\frac{\left(x-1\right)^2}{2}\)
\(=\frac{\left(x-\sqrt{x}-2\right)-\left(x+\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}.\frac{\left(x-1\right)^2}{2}\)
\(=\frac{-2\sqrt{x}}{\left(x-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(x-1\right)^2}{2}\)\(=\frac{-\sqrt{x}}{\sqrt{x}+1}.\left(x-1\right)=\frac{-x\sqrt{x}+\sqrt{x}}{\sqrt{x}+1}\)
7 tháng 6 2017
\(x^4+9=5x\left(x^2-3\right)\)
\(\Leftrightarrow x^4-5x^3+15x+9=0\)
\(\Leftrightarrow\left(x^4-3x^3\right)-\left(2x^3-6x^2\right)-\left(6x^2-18x\right)-\left(3x-9\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x^3-2x^2-6x-3\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left[\left(x^3+x^2\right)-\left(3x^2+3x\right)-\left(3x+3\right)\right]=0\)
\(\Leftrightarrow\left(x-3\right)\left(x+1\right)\left(x^2-3x-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=0\\x+1=0\\x^2-3x-3=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x-3=0\\x+1=0\\\Delta=\left(-3\right)^2-4.\left(-3\right)=21>0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-1\\x=\dfrac{3+\sqrt{21}}{2}\\x=\dfrac{3-\sqrt{21}}{2}\end{matrix}\right.\)
Vậy phương trình đã cho có \(S=\left\{3;-1;\dfrac{3+\sqrt{21}}{2};\dfrac{3-\sqrt{21}}{2}\right\}\)
x^4-4x^2-4-8x^2+16\(\sqrt{2}\)x-8=0
<=> (x^2-2)^2 -8( x-\(\sqrt{2}\))^2=0
<=> (x-\(\sqrt{2}\))^2( (x+\(\sqrt{2}\))^2-8))=0
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