Quadratic Inequalities

  1. Solving Quadratic Equations

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

              Definition of the Discriminant

    We define the discriminant  of the quadratic 

              ax2 + bx + c 

    as

              D = b2 - 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

    How many roots does

            1045456564x2 + 3x + 2134534265256 

    have?


    Solution

    It is clear that 4ac is larger than b2 = 9.

    Hence

            D = 9 - 4ac < 0

    So that the quadratic has no real roots.


  2. Quadratic Inequalities


    Example:

    Solve

            x2 - x - 6 > 0

    Solution:  

    First we solve the equality by factoring:

            (x - 3)(x + 2) = 0

    Hence 

            x     = -2 or x = 3

    Next we cut the number line into three regions: 

             x < -2,    -2 < x < 3,    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 or x > 3

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

            y = x2 - x - 6 

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

     



  3. Applications

    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: 

                Area of the walk = p(4 + r)2 - p( r)2 = .44(p)( r)2

    Dividing by p we have,

            (4 + r)2 - r2 = .44r2  

    multiplying out, we get,

            16 + 8r + r2 -r2 = .44r2  

    or

            .44r2 -8r -16

    Now use the quadratic formula:

            a = .44, b = -8, c = -16

    so

            r = 1.1 or r = -.1

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

    Example: 

    The profit function for burgers at Heavenly is given by

            P = 35x - x2/25,000,000 - 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?

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