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Mathematics LibreTexts

2.4: Quadratic Inequalities

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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

    Example 1

    How many roots does

    \[1045456564x^2 + 3x + 2134534265256\]

    have?

    Solution

    It is clear that \(4ac\) is larger than \(b^2= 9\). Hence

    \[D = 9 - 4ac < 0\]

    so that the quadratic has no real roots.

    Quadratic Inequalities

    Example 2

    Solve

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

    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

    Example 3

    A 4 ft walkway surrounds a circular flower garden, as shown in the sketch. The area of the walk is 44% of the area of the garden. Find the radius of the garden.

    Solution:

    \[\begin{align} \text{Area of walk} &= p(4+r)^2-p(r)^2 \\ &= .44(p)(r)^2 \end{align}\]

    Dividing by \(p\) we have,

    \[(4 + r)^2- r^2 = .44r^2\]

    multiplying out, we get,

    \[16 + 8r + r^2 -r^2 = .44r^2\]

    or

    \[.44r^2-8r -16.\]

    Now use the quadratic formula:

    \[a = .44, b = -8, c = -16\]

    so

    \[r = 1.1 \;\;\; \text{or} \;\;\; r = -.1.\]

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

    Exercise

    The profit function for burgers at Heavenly is given by

    \[P = 35x - \dfrac{x^2}{25,000,000} - 40,000.\]

    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.