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Thread: finding common roots

  1. #1
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    finding common roots

    please help me with this question

    for which real values of b do the equations

    x^3 + bx^2 + 2bx -1 =0 and x^2 + (b-1)x + b = 0

    have a common root

    thank you
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let $\displaystyle x_0$ be the common root.
    $\displaystyle \displaystyle\left\{\begin{array}{cc}x_0^3+bx_0^2+ 2bx_0-1=0 & (1)\\
    x_0^2+(b-1)x_0+b=0 & (2)\end{array}\right.$
    We observe that $\displaystyle x_0\neq 0$.
    Multiplying (2) by $\displaystyle x_0$ and then substracting from (1), we have
    $\displaystyle x_0^2+bx_0-1=0\Rightarrow x_0^2+bx_0=1$ (3)
    The equality (2) can be written as $\displaystyle x_0^2+bx_0-x_0+b=0$
    Using (3) we have $\displaystyle 1-x_0+b=0\Rightarrow x_0=b+1$.
    Plug $\displaystyle x_0$ in (2) and we have $\displaystyle 2b^2+3b=0\Rightarrow b_1=0, \ b_2=\displaystyle-\frac{3}{2}$.
    If $\displaystyle b=0$, the common root is $\displaystyle x_0=1$.
    If $\displaystyle \displaystyle b=-\frac{3}{2}$, the common root is $\displaystyle \displaystyle x_0=-\frac{1}{2}$.
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