Cho a/b = b/c = c/a hãy tính giá trị biểu thức M = (a^2 + b^2 + c^2 ) / ( a+b+c )^2

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Cho a/b = b/c = c/a hãy tính giá trị biểu thức M = (a^2 + b^2 + c^2 ) / ( a+b+c )^2

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Kaylee 2 tuần 2021-08-28T07:49:18+00:00 1 Answers 0 views 0

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    2021-08-28T07:50:39+00:00

    Đáp án:  Đặt

    $\begin{array}{l}
    \frac{a}{b} = \frac{b}{c} = \frac{c}{a} = k \Rightarrow \left\{ \begin{array}{l}
    a = b.k\\
    b = c.k\\
    c = a.k
    \end{array} \right. \Rightarrow \left\{ \begin{array}{l}
    a = \left( {c.k} \right).k = c.{k^2}\\
    b = c.k
    \end{array} \right.\\
     \Rightarrow M = \frac{{\left( {{a^2} + {b^2} + {c^2}} \right)}}{{{{\left( {a + b + c} \right)}^2}}}\\
     = \frac{{{{\left( {c.{k^2}} \right)}^2} + {c^2}.{k^2} + {c^2}}}{{{{\left( {c.{k^2} + c.k + c} \right)}^2}}}\\
     = \frac{{{c^2}.{k^4} + {c^2}.{k^2} + {c^2}}}{{{c^2}.{{\left( {{k^2} + k + 1} \right)}^2}}}\\
     = \frac{{{c^2}\left( {{k^4} + {k^2} + 1} \right)}}{{{c^2}{{\left( {{k^2} + k + 1} \right)}^2}}}\\
     = \frac{{{k^4} + {k^2} + 1}}{{{{\left( {{k^2} + k + 1} \right)}^2}}}
    \end{array}$

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