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Math Help - Quadratic Functions and Equations

  1. #1
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    Quadratic Functions and Equations

    I was assigned an achievement check a few days back and it is due soon.

    Question: Find k such that the graph of y=9x^2 + 3kx + k:
    a) intersects the x-axis in one point only
    b) intersects the x-axis in two points
    c) does not intersect the x-asix

    I am unable to figure out how to solve it. Can someone explain to me how to do this please?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by DFKnight View Post
    I was assigned an achievement check a few days back and it is due soon.

    Question: Find k such that the graph of y=9x^2 + 3kx + k:
    a) intersects the x-axis in one point only
    b) intersects the x-axis in two points
    c) does not intersect the x-asix

    I am unable to figure out how to solve it. Can someone explain to me how to do this please?
    for x-intercept, y = 0. so we are concerned with the condition:
    9x^2 + 3kx + k = 0

    by the quadratic formula:
    x = [-3k +/- sqrt(9K^2 - 4(9)k)]/2(9)
    => x = [-3k +/- sqrt(9K^2 - 36k)]/18

    a) we have one x-intercept if 9K^2 - 36k = 0
    => 9k(k - 4) = 0
    => k = 0 or k = 4 for 1 x-intercept

    b) for 2 x-intercepts, 9K^2 - 36k > 0
    => 9k(k - 4) > 0
    => k < 0 or k > 4
    so to satisfy this condition, k has to be on the interval (-infinity, 0) U (4, infinity)

    we had to test these values on a number line to know which way the inequalities should go

    c) for no x-intercept, 9K^2 - 36k < 0
    => 9k(k - 4) < 0
    => 0 < k < 4

    Do you know why these conditions have to be fulfilled the way they are?
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  3. #3
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    Thanks a lot for the detailed explaination. And yes, I do understand why these conditions have to be fulfilled the way they are.
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