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2.4: Quadratic Inequalities

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    233
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    Solving Quadratic Equations

    We solve quadratic equations by either factoring or using the quadratic formula.

    Definition: The Discriminant

    We define the discriminant of the quadratic

    \[ax^2 + bx + c \]

    as

    \[D = b^2 - 4ac.\]

    The discriminant is the number under the square root in the quadratic formula. We immediately get

    D # of Roots
    > 0 2
    < 0 0
    0 1

    so that the quadratic has no real roots.

    Quadratic Inequalities

    \[x^2- x - 6 > 0\]

    Solution:

    First we solve the equality by factoring:

    \[(x - 3)(x + 2) = 0\]

    hence

    \[x = -2 \; \text{ or } \; x = 3.\]

    Next we cut the number line into three regions:

    \[x < -2, -2 < x < 3, \text{ and } x > 3.\]

    On the first region (test \(x = -3\)), the quadratic is positive, on the second region (test \(x = 0\)) the quadratic is negative, and on the third region (test \(x = 5\)) the quadratic is positive.

    Region Test Value y-Value Sign
    \(x < 2\) \(x = -3\) \(y = 6\) \(+\)
    \(-2 < x < 3\) \(x = 0\) \(y = -6\) \(-\)
    \(x > 3\) \(x = 5\) \(y = 14\) \(+\)

    We are after the positive values since the equation is "\(> 0\)". Hence our solution is region 1 and region 2:

    \[x < -2 \; \text{ or } \; x > 3.\]

    We will see how to verify this on a graphing calculator by noticing that

    \[y = x^2 - x - 6 \]

    stays above the x-axis when \(x < -2\) and when \(x > 3\).

    Applications

    since -.1 does not make sense, we can say that the radius of the garden is 1.1 feet.

    Where \(x\) represents the number of skiers that come on a given day. How many skiers paying for Heavenly will produce the maximal profit?

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.


    This page titled 2.4: Quadratic Inequalities is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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