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Math Help - discriminant with quadratic inequalities

  1. #1
    Newbie Tom G's Avatar
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    discriminant with quadratic inequalities

    Need help with this questions as soon as possible please.

    1. find the range of values that b can take if x^2 + 5bx + b is positive for all real x.
    2. use the discriminant to find the range of values k can take for kx^5 + 3x + k -4 = 0 to have two real distinct roots.
    Last edited by Tom G; December 13th 2006 at 11:20 AM.
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  2. #2
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    Hello, Tom G!

    Find the range of values that b can take
    if x^2 + 5bx + b is positive for all real x.

    The function y \:=\:x^2 + 5bx + b is an up-opening parabola.

    Its vertex is at: . x \:=\:\frac{-5b}{2}
    . . and: y \:=\:\left(-\frac{5b}{2}\right)^2 + 5b\left(-\frac{5b}{2}\right) + b\:=\:\frac{b(4-25b)}{4}

    Hence, its vertex (lowest point) is: . \left(-\frac{5b}{2},\:\frac{b(4-25b)}{4}\right)


    If the parabola is to be above the x-axis, then: . \frac{b(4-25b)}{4} \:>\:0


    There are two cases:

    [1]\;\;b < 0 and 4 - 25b \,< \,0
    . . .But this gives us: . b < 0 and b > \frac{4}{25} . . . impossible

    [2]\;\;b > 0 and 4 - 25b \,> \,0
    . . .which gives us: . b > 0 and b < \frac{4}{25}

    Therefore, the range is: . \boxed{0 \,< \,b \,< \,\frac{4}{25}}

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  3. #3
    Member Glaysher's Avatar
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    Quote Originally Posted by Tom G View Post
    Need help with this questions as soon as possible please.

    1. find the range of values that b can take if x^2 + 5bx + b is positive for all real x.
    2. use the discriminant to find the range of values k can take for kx^5 + 3x + k -4 = 0 to have two real distinct roots.
    2.

    For the quadratic to have two distinct real roots the discriminant b^2 - 4ac > 0

    (the other two possibilities are equal roots b^2 - 4ac = 0
    or no real roots b^2 - 4ac < 0

    So a = k, b = 3 and c = k - 4

    So get 3^2 - 4k(k - 4) > 0

    9 - 4k^2 + 16k > 0

    4k^2 - 16k - 9 < 0 by rearranging

    (2k + 1)(2k - 9) < 0

    See image below for a sketch

    The part of the graph that has been marked is below the k-axis as this corresponds to where 4k^2 - 16k - 9 is negative or less than zero

    So the answer is:

    - \frac{1}{2} < k < \frac{9}{2}
    Attached Thumbnails Attached Thumbnails discriminant with quadratic inequalities-quadratic.jpg  
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