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Math Help - Fantastic system

  1. #1
    Super Member dhiab's Avatar
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    Fantastic system

    Find all reals numbers solutions of system :
    \left\{ \begin{array}{l}<br />
{\rm a + b + c = 4} \\ <br />
{\rm a}^{\rm 2} + b^2 + c^2 = 14 \\ <br />
a^3 + b^3 + c^3 = 34 \\ <br />
\end{array} \right.<br />
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  2. #2
    MHF Contributor red_dog's Avatar
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    a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)\Rightarrow ab+ac+bc=1

    a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc\Rightarrow abc=-6

    So we have

    \left\{\begin{array}{ll}a+b+c=4\\ab+ac+bc=1\\abc=-6\end{array}\right.

    Then a,b,c are the roots of the equation x^3-4x^2+x+6=0

    The equation has the solutions x_1=-1, \ x_2=2, \ x_3=3

    Then \left\{\begin{array}{ll}a=-1\\b=2\\c=3\end{array}\right. or \left\{\begin{array}{ll}a=2\\b=-1\\c=3\end{array}\right. or.....etc.
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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by red_dog View Post
    a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)\Rightarrow ab+ac+bc=1

    a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc\Rightarrow abc=-6

    So we have

    \left\{\begin{array}{ll}a+b+c=4\\ab+ac+bc=1\\abc=-6\end{array}\right.

    Then a,b,c are the roots of the equation x^3-4x^2+x+6=0

    The equation has the solutions x_1=-1, \ x_2=2, \ x_3=3

    Then \left\{\begin{array}{ll}a=-1\\b=2\\c=3\end{array}\right. or \left\{\begin{array}{ll}a=2\\b=-1\\c=3\end{array}\right. or.....etc.
    Thank you
    The system has the solutions
     <br />
\begin{array}{l}<br />
\left\{ \begin{array}{l}<br />
a = 2 \\ <br />
b = 3 \\ <br />
c = - 1 \\ <br />
\end{array} \right.\left\{ \begin{array}{l}<br />
a = 3 \\ <br />
b = - 1 \\ <br />
c = 2 \\ <br />
\end{array} \right.\left\{ \begin{array}{l}<br />
a = 3 \\ <br />
b = 2 \\ <br />
c = - 1 \\ <br />
\end{array} \right. \\ <br />
\\ <br />
\end{array}<br /> <br />
    And :
    \left\{ \begin{array}{l}<br />
a = - 1 \\ <br />
b = 2 \\ <br />
c = 3 \\ <br />
\end{array} \right.\left\{ \begin{array}{l}<br />
a = - 1 \\ <br />
b = 3 \\ <br />
c = 2 \\ <br />
\end{array} \right.\left\{ \begin{array}{l}<br />
a = 2 \\ <br />
b = - 1 \\ <br />
c = 3 \\ <br />
\end{array} \right.
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