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Finding m in quadratic equation 12x^2 + 8mx + (4m-3) = 0

  1. Apr 5, 2006 #1

    danago

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    Gold Member

    Hey. Today i had a test on quadratics and discriminants. I think i did fairly well, but i am a bit confused about one of the questions i had in it.

    We were given the following quadratic equation:

    [tex]12x^2 + 8mx + (4m-3) = 0[/tex]

    What we had to do was prove that for any integer value of m, the equation would have rational solutions.

    What i did was first take the discriminant of the equation:
    [tex]\Delta = (8m)^2 - 4(12)(4m-3)[/tex]

    simplified it:
    [tex]\Delta = 64m^2 - 192m + 144[/tex]

    From that, i then created a table of values. I made a table of values for m (-5 to 5), the discriminant, then the square root of the discriminant. Since all of the square roots were whole numbers, i could have used that as a reason why all the solutions would be rational, but its not really proving that all values for m will follow the rule. It just shows that 10 of my chosen values work.

    from here i wasnt really sure what to do. I noted that the discriminant of the original equation was a quadratic function itself, so i graphed it, and noticed that for every integer value of x, its corrosponding y value will be a perfect square number. I wrote about this observation, and am just hoping its close enough to what i should have done.

    If anybody has any idea about if i should have gone about this another way, or if i was right, please post :) all comments greatly appreciated.

    Thanks,
    Dan.
     
  2. jcsd
  3. Apr 5, 2006 #2
    Try factoring the discriminant. Then use the quadratic formula to show that all answers are rational.
     
  4. Apr 5, 2006 #3

    VietDao29

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    Homework Helper

    So far so good.
    Do you know what are rational numbers? They are numbers that can be expressed in a form of a fraction p / q, where p, and q are whole numbers (integers).
    To prove that the solutions are rational for any integer value of m, you should prove that:
    [tex]\frac{-8m \pm \sqrt{\Delta}}{24}[/tex] is a rational number, right?
    Note that:
    [tex]\Delta = 64m ^ 2 - 192m + 144 = (8m - 12) ^ 2[/tex].
    Can you go from here? :)
     
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