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Thread: Finding for interval of m in quadratic equation.

  1. August 23rd 2011, 10:44 PM #1
    sumedh
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    Finding for interval of m in quadratic equation.

    Find the interval in which 'm' lies so that the expression
    \frac{mx^2+3x-4}{-4x^2+3x+m}
    can take all real values ,where x is real.


    i have equated this equation to y
    y=\frac{mx^2+3x-4}{-4x^2+3x+m}

    then
    (-4y-m)x^2+(3y-3)x+(ym-4)=0
    but then it becomes quadratic equation in many variables .


    please give hint or steps to solve the problem
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  2. August 23rd 2011, 11:02 PM #2
    FernandoRevilla
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    Re: Finding for interval of m in quadratic equation.

    For all x,m\in \mathbb{R} , numerator and denominator belong to \mathbb{R} . You only need to study for which values of m there exists x such that -4x^2+3x+m=0 . For those values of m the quotient is not defined for all x\in\mathbb{R} .
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  3. August 24th 2011, 12:22 AM #3
    sumedh
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    Re: Finding for interval of m in quadratic equation.

    Quote Originally Posted by FernandoRevilla View Post
    . You only need to study for which values of m there exists x such that -4x^2+3x+m=0 .

    how to find m
    should i equate D=0 or -D/4a=0???
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  4. August 24th 2011, 12:36 AM #4
    FernandoRevilla
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    Re: Finding for interval of m in quadratic equation.

    The roots of -4x^2+3x+m=0 are x_1=\frac{3+\sqrt{9+16m}}{8} and x_2=\frac{3-\sqrt{9+16m}}{8} .
    Last edited by FernandoRevilla; August 24th 2011 at 06:59 AM.
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  5. August 24th 2011, 02:08 AM #5
    sumedh
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    Re: Finding for interval of m in quadratic equation.

    the final answer is m \epsilon[1,7]
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  6. August 24th 2011, 06:57 AM #6
    FernandoRevilla
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    Re: Finding for interval of m in quadratic equation.

    Quote Originally Posted by sumedh View Post
    the final answer is m \epsilon[1,7]
    m<-9/16 .
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  7. August 25th 2011, 10:25 AM #7
    sumedh
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    Re: Finding for interval of m in quadratic equation.

    Then the question might be wrong

    Thank you very much for your help and the valuable time you spared for me.
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