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