\({\left( {3 + \sqrt 5 } \right)^x} + {\left( {3 - \sqrt 5 } \right)^x} = {3.2^x} \Leftrightarrow {\left( {\frac{{3 + \sqrt 5 }}{2}} \right)^x} + {\left( {\frac{{3 - \sqrt 5 }}{2}} \right)^x} = 3\)
Đặt \(t = {\left( {\frac{{3 + \sqrt 5 }}{2}} \right)^x},t > 0 \Rightarrow \left( {\frac{{3 - \sqrt 5 }}{2}} \right) = \frac{1}{t} \Rightarrow PT \Leftrightarrow t + \frac{1}{t} = 3 \Leftrightarrow {t^2} - 3t + 1 = 0\) \( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{t = \frac{{3 - \sqrt 5 }}{2}}\\{t = \frac{{3 + \sqrt 5 }}{2}}\end{array}} \right.\)
\( \Rightarrow \left[ {\begin{array}{*{20}{c}}{{{\left( {\frac{{3 + \sqrt 5 }}{2}} \right)}^x} = \frac{{3 - \sqrt 5 }}{2}}\\{{{\left( {\frac{{3 + \sqrt 5 }}{2}} \right)}^x} = \frac{{3 + \sqrt 5 }}{2}}\end{array} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{{\left( {\frac{{3 + \sqrt 5 }}{2}} \right)}^x} = {{\left( {\frac{{3 + \sqrt 5 }}{2}} \right)}^{ - 1}}}\\{{{\left( {\frac{{3 + \sqrt 5 }}{2}} \right)}^x} = \frac{{3 + \sqrt 5 }}{2}}\end{array}} \right.} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = - 1}\\{x = 1}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{{x_1} = - 1}\\{{x_2} = 1}\end{array} \Rightarrow A = 2} \right.\)
