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Math Help - Algeb

  1. #1
    Member srirahulan's Avatar
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    Post Algeb

    If x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0 quadratic equations have a common root.prove b=c or the other roots are the roots of a(b+c)x^2+(b+c)-abc=0
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  2. #2
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    Re: Algeb

    Quote Originally Posted by srirahulan View Post
    If x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0 quadratic equations have a common root.prove b=c or the other roots are the roots of a(b+c)x^2+(b+c)-abc=0
    The roots of the first equation are

    \displaystyle \begin{align*} x &= \frac{-ab \pm \sqrt{(ab)^2 - 4(1)(c)}}{2(1)} \\ &= \frac{-ab \pm \sqrt{a^2b^2 - 4c}}{2} \end{align*}

    and the roots of the second equation are

    \displaystyle \begin{align*} x &= \frac{-ac \pm \sqrt{(ac)^2 - 4(1)(b)}}{2(1)} \\ &= \frac{-ac \pm \sqrt{a^2c^2 - 4b}}{2} \end{align*}

    Since they share a root, there is some value such that

    \displaystyle \begin{align*} \frac{-ab \pm \sqrt{a^2b^2 - 4c}}{2} &= \frac{-ac \pm \sqrt{a^2c^2 - 4b}}{2} \\ -ab \pm \sqrt{a^2b^2 - 4c} &= -ac \pm \sqrt{a^2c^2 - 4b} \end{align*}

    If \displaystyle \begin{align*} b = c \end{align*}, we have

    \displaystyle \begin{align*} LHS &= -ab \pm \sqrt{a^2b^2 - 4c} \\ &= -ab \pm \sqrt{a^2b^2 - 4b} \\ \\ RHS &= -ac \pm \sqrt{a^2c^2 - 4b} \\ &= -ab \pm \sqrt{a^2b^2 - 4b} \\ &= LHS \end{align*}

    So that means for the two equations to share a root, it's possible that \displaystyle \begin{align*} b = c \end{align*}.
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  3. #3
    Member srirahulan's Avatar
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    Re: Algeb

    I can't understand you statements.pls reply,
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  4. #4
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    Re: Algeb

    Quote Originally Posted by srirahulan View Post
    ....are the roots of a(b+c)x^2+(b+c)-abc=0
    k = [abc - (b+c)] / [a(b+c)]
    x = +- SQRT(k)
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  5. #5
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    Re: Algeb

    Quote Originally Posted by srirahulan View Post
    If x^2+abx+c=0 \mbox\ {and} \ x^2+acx+b=0 quadratic equations have a common root.prove b=c or the other roots are the roots of a(b+c)x^2+(b+c)-abc=0
    x^2 + abx + c = 0 [1]
    x^2 + acx + b = 0 [2]

    Roots of [1] = u,v
    Roots of [2] = u,w
    Then:
    ab = u + v
    c = uv : v = c/u
    ab = u + c/u : a = (u + c/u) / b [3]

    Similarly: a = (u + b/u) / c [4]

    [3][4]: (u + c/u) / b = (u + b/u) / c
    Simplify:
    bu^2 + b^2 = cu^2 + c^2
    b = c
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