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Câu hỏi: Biết rằng phương trình ${{3}^{x}}-{{5.3}^{\dfrac{x}{2}}}+4=0$ có đúng hai nghiệm thực phân biệt ${{x}_{1}},{{x}_{2}}.$ Tính giá trị của biểu thức $S=\left| {{x}_{1}}-{{x}_{2}} \right|.$
A. $S=4{{\log }_{3}}2.$
B. $S=2{{\log }_{3}}\dfrac{4}{3}.$
C. $S=6{{\log }_{3}}2.$
D. $S=4{{\log }_{3}}\dfrac{4}{3}.$
A. $S=4{{\log }_{3}}2.$
B. $S=2{{\log }_{3}}\dfrac{4}{3}.$
C. $S=6{{\log }_{3}}2.$
D. $S=4{{\log }_{3}}\dfrac{4}{3}.$
Ta có ${{3}^{x}}-{{5.3}^{\dfrac{x}{2}}}+4=0\Leftrightarrow {{\left( {{3}^{\dfrac{x}{2}}} \right)}^{2}}-{{5.3}^{\dfrac{x}{2}}}+4=0\Leftrightarrow \left[ \begin{array}{*{35}{l}}
{{3}^{\dfrac{x}{2}}}=1 \\
{{3}^{\dfrac{x}{2}}}=4 \\
\end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{*{35}{l}}
\dfrac{x}{2}=0 \\
\dfrac{x}{2}={{\log }_{3}}4 \\
\end{array} \right.\Leftrightarrow \left[ \begin{array}{*{35}{l}}
x=0 \\
x=2{{\log }_{3}}4 \\
\end{array} \right.\Rightarrow S=2{{\log }_{3}}4=4{{\log }_{3}}2.$
{{3}^{\dfrac{x}{2}}}=1 \\
{{3}^{\dfrac{x}{2}}}=4 \\
\end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{*{35}{l}}
\dfrac{x}{2}=0 \\
\dfrac{x}{2}={{\log }_{3}}4 \\
\end{array} \right.\Leftrightarrow \left[ \begin{array}{*{35}{l}}
x=0 \\
x=2{{\log }_{3}}4 \\
\end{array} \right.\Rightarrow S=2{{\log }_{3}}4=4{{\log }_{3}}2.$
Đáp án A.