Giải pt ((x+3)/(x-5))^2+6((x-3)/(x+5))^2=7((x^2-9)/(x^2-25))

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1. Đáp án:

   \[\left[ \begin{array}{l}
   x = 0\\
   x = \frac{{28 \pm \sqrt {409} }}{5}
   \end{array} \right.\]

   Giải thích các bước giải:

    Ta có:

   \(\begin{array}{l}
   {\left( {\frac{{x + 3}}{{x – 5}}} \right)^2} + 6.{\left( {\frac{{x – 3}}{{x + 5}}} \right)^2} = 7.\frac{{{x^2} – 9}}{{{x^2} – 25}}\\
    \Leftrightarrow {\left( {\frac{{x + 3}}{{x – 5}}} \right)^2} – 7.\frac{{\left( {x – 3} \right)\left( {x + 3} \right)}}{{\left( {x – 5} \right)\left( {x + 5} \right)}} + 6.{\left( {\frac{{x – 3}}{{x + 5}}} \right)^2} = 0\\
    \Leftrightarrow {\left( {\frac{{x + 3}}{{x – 5}}} \right)^2} – 7.\frac{{x + 3}}{{x – 5}}.\frac{{x – 3}}{{x + 5}} + 6{\left( {\frac{{x – 3}}{{x + 5}}} \right)^2} = 0\\
    \Leftrightarrow \left( {\frac{{x + 3}}{{x – 5}} – \frac{{x – 3}}{{x + 5}}} \right)\left( {\frac{{x + 3}}{{x – 5}} – 6.\frac{{x – 3}}{{x + 5}}} \right) = 0\\
    \Leftrightarrow \left[ \begin{array}{l}
   \frac{{x + 3}}{{x – 5}} = \frac{{x – 3}}{{x + 5}}\\
   \frac{{x + 3}}{{x – 5}} = 6.\frac{{x – 3}}{{x + 5}}
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   {x^2} + 8x + 15 = {x^2} – 8x + 15\\
   {x^2} + 8x + 15 = 6.\left( {{x^2} – 8x + 15} \right)
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x = 0\\
   5{x^2} – 56x + 75 = 0
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x = 0\\
   x = \frac{{28 \pm \sqrt {409} }}{5}
   \end{array} \right.
   \end{array}\)

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