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Math Help - Four roots of an equation

  1. #1
    Junior Member
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    Aug 2008
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    Four roots of an equation

    Greetings,
    This is a homework. Not quite an urgent one - I need it for monday and I have had it since monday. Unfortunately, few ideas occured to me.
    Obviously, a reasonable step, perhaps, is to make a substitution, but then, I believe, we get only a link between m and y, assuming y is the substitution y = x*x -2x + 2, and no link between m and x.
    The problem is taken from a university entrance test. We need this equation to have 4 different roots.
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  2. #2
    MHF Contributor
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    Substitution is a good idea. You will get y^2 - 3my + 3 = 0, from which you can use the quadratic formula:

    y = \frac{3m \pm \sqrt{(3m)^2 - 4(1)(3)}}{2}

    y = \frac{3m \pm \sqrt{9m^2 - 12}}{2}

    If this equation is to have any real roots at all, 9m^2 - 12 \geq 0

    9m^2 \geq 12

    m^2 \geq \frac{4}{3}

    Then you have x^2 + 2x + 2 = \frac{3m \pm \sqrt{9m^2 - 12}}{2}

    x^2 + 2x + 2 - \frac{3m \pm \sqrt{9m^2 - 12}}{2} = 0

    and you can use the quadratic formula again.
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