Solving Quadratics

  1. Factoring

    Zero Product Theorem  If the product of two factors is zero then at least one of the two factors must be zero.  

     

    Example

            x(x - 1) = 0 

    implies that either 

            x = 0     or     x = 1 

     

    A Review of Factoring  

    Done by working several examples such as

            x2 - 3x - 10 = (x - 5)(x + 2)

    For a complete review of factoring go to Factoring and for an interactive tutorial on the AC method for factoring go to http://mathcsjava.emporia.edu/~greenlar/AC/AC.html

     

    The Square Root Method  

    If the middle term of a quadratic is zero then the best way to solve is to isolate the x2 term and then take square roots of both sides.  

    Example

            4x2 - 9 = 0         Subtract 9 from both sides to produce

            4x2 = 9               Divide by 4

            x2  = 9/4             take the square root

            x = +- 3/2 

    Caution:  Don't forget that you get two solutions- the "plus" and the "minus" solution.


  2. Completing the Square  

    For a full review of completing the square go to Completing the Square for an interactive tutorial on completing the square go to http://mathcsjava.emporia.edu/~greenlar/CompleteTheSquare/CompleteTheSquare.html

    Here is a brief explanation by example: 

            2x2 - 8x + 2

     

    1. Factor the leading coefficient:  2(x2 - 4x + 1)

    2. Calculate -b/2:  -(-4)/2 = 2

    3. Square the solution above:  22  = 4

    4. Add and subtract answer from part three inside parentheses: 
              2(x2 - 4x + 4 - 4 + 1) 

    5. Regroup:  2[(x2 - 4x + 4 ) - 3]

    6. Factor the inner parentheses using part two as a hint:  2[(x - 2)2 - 3]

    7. Multiply out the outer constant:    2(x - 2)2 - 6

    8. Breath a sigh of relief.

     

  3. Quadratic Formula

    Recall (otherwise memorize) the quadratic formula: The solutions to 

            ax2  + bx + c = 0 

    are

           

    and 

           

    We will use this often, so memorize it.  Recall that the discriminant

            D = b2  - 4ac

    is a convenient measure of determining how many roots (solutions) there are.  We have:

     
    D Number of Roots
    Positive 2
    Negative 0
    zero 1

     

    Example  

    The quadratic

            3245234543x2 - 432523465236432x - 4598674689

    has two roots since D is positive.

    For more information on the quadratic formula go to The Quadratic Formula

 


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Questions, Comments and Suggestions:  Email:  greenl@ltcc.edu