Giải phương trình: \(2\sqrt[4]{\frac{x^2}{3}+4}=1+\sqrt{\frac{3x}{2}}\)
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24 tháng 7 2017
a/ \(\frac{1}{2-\sqrt{3}}+\frac{3+\sqrt{3}}{\sqrt{3}}-\frac{4}{\sqrt{3}-1}\)
\(=2+\sqrt{3}+\sqrt{3}+1-2\sqrt{3}-2\)
\(=1\)
b/ \(\sqrt{3x+40}-4=x\)
\(\sqrt{3x+40}=x+4\)
Điều kiện: \(\hept{\begin{cases}3x+40\ge0\\x+4\ge0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x\ge-\frac{40}{3}\\x\ge-4\end{cases}}\)
\(\Leftrightarrow x\ge-\frac{40}{3}\)
Ta có: \(3x+40=x^2+8x+16\)
\(\Leftrightarrow x^2+5x-24=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=-8\left(l\right)\\x=3\end{cases}}\)
23 tháng 7 2017
a. Ta có \(\frac{1}{2-\sqrt{3}}+\frac{3\sqrt{3}}{\sqrt{3}}-\frac{4}{\sqrt{3}-1}=\frac{2+\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}+3-\frac{4\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{2+\sqrt{3}}{4-3}+3-\frac{4\left(\sqrt{3}+1\right)}{3-1}=2+\sqrt{3}+3-2\sqrt{3}-2=3-\sqrt{3}\)
b. \(\sqrt{3x+40}-4=x\)
ĐK \(3x+40\ge0\Leftrightarrow x\ge-\frac{40}{3}\)
\(\Leftrightarrow\sqrt{3x+40}=x+4\)\(\Leftrightarrow\hept{\begin{cases}x\ge-4\\3x+40=x^2+8x+16\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge-4\\x^2+5x-24=0\end{cases}}}\)
\(\Leftrightarrow\hept{\begin{cases}x\ge-4\\\left(x+8\right)\left(x-3\right)=0\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge-4\\x=-8;x=3\end{cases}}}\Leftrightarrow x=3\left(tm\right)\)
Vậy x=3
10 tháng 8 2017
\(\sqrt{3x^2-5x+1}-\sqrt{x^2-2}=\sqrt{3\left(x^2-x-1\right)}-\sqrt{x^2-3x+4}\)
TN
4 tháng 7 2017
b)\(\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2x-\frac{5}{x}}\)
\(pt\Leftrightarrow\frac{4}{x}+\sqrt{x-\frac{1}{x}}-\sqrt{\frac{3}{2}}=x+\sqrt{2x-\frac{5}{x}}-\sqrt{\frac{3}{2}}\)
\(\Leftrightarrow\left(\frac{4}{x}-x\right)+\frac{x-\frac{1}{x}-\frac{3}{2}}{\sqrt{x-\frac{1}{x}}+\sqrt{\frac{3}{2}}}=\frac{2x-\frac{5}{x}-\frac{3}{2}}{\sqrt{2x-\frac{5}{x}}+\sqrt{\frac{3}{2}}}\)
\(\Leftrightarrow\frac{-\left(x-2\right)\left(x+2\right)}{x}+\frac{\frac{\left(x-2\right)\left(2x+1\right)}{2x}}{\sqrt{x-\frac{1}{x}}+\sqrt{\frac{3}{2}}}-\frac{\frac{\left(x-2\right)\left(4x+5\right)}{2x}}{\sqrt{2x-\frac{5}{x}}+\sqrt{\frac{3}{2}}}=0\)
\(\Leftrightarrow\left(x-2\right)\left(\frac{-\left(x+2\right)}{x}+\frac{\frac{\left(2x+1\right)}{2x}}{\sqrt{x-\frac{1}{x}}+\sqrt{\frac{3}{2}}}-\frac{\frac{\left(4x+5\right)}{2x}}{\sqrt{2x-\frac{5}{x}}+\sqrt{\frac{3}{2}}}\right)=0\)
Pt trong ngoặc VN suy ra x=2
TN
4 tháng 7 2017
a)\(x^2+3\sqrt{x^2-1}=\sqrt{x^4-x^2+1}\)
\(\Leftrightarrow x^2+3\sqrt{x^2-1}-1=\sqrt{x^4-x^2+1}-1\)
\(\Leftrightarrow\frac{x^2\left(3\sqrt{x^2-1}+1\right)}{3\sqrt{x^2-1}+1}+\frac{9\left(x^2-1\right)-1}{3\sqrt{x^2-1}+1}=\frac{x^4-x^2+1-1}{\sqrt{x^4-x^2+1}+1}\)
\(\Leftrightarrow\frac{9x^2-10+3x^2\sqrt{x^2-1}+x^2}{3\sqrt{x^2-1}+1}=\frac{x^4-x^2}{\sqrt{x^4-x^2+1}+1}\)
\(\Leftrightarrow\frac{\sqrt{x^2-1}\left(3x^2+10\sqrt{x^2-1}\right)}{3\sqrt{x^2-1}+1}=\frac{x^2\left(x-1\right)\left(x+1\right)}{\sqrt{x^4-x^2+1}+1}\)
\(\Leftrightarrow\frac{\sqrt{\left(x-1\right)\left(x+1\right)}\left(3x^2+10\sqrt{x^2-1}\right)}{3\sqrt{x^2-1}+1}-\frac{x^2\left(x-1\right)\left(x+1\right)}{\sqrt{x^4-x^2+1}+1}=0\)
\(\Leftrightarrow\left(x-1\right)\left(x+1\right)\left(\frac{\frac{1}{\sqrt{x^2-1}}\left(3x^2+10\sqrt{x^2-1}\right)}{3\sqrt{x^2-1}+1}-\frac{x^2}{\sqrt{x^4-x^2+1}+1}\right)=0\)
pt trong căn vô nghiệm
suy ra x=1; x=-1