Solving Quadratic (Degree 2) Equations 
                       
                      A Quadratic Equation is an equation that can be written in the following form, called standard form: 
                        
                                                                        ax2+bx+c=0   
                                                                               
                      The coefficients a, b, and c are real numbers with a ≠ 0.   
                       
                                                                               
                                                                               
                      When solving a quadratic equation with both b and c ≠ 0 one of the two following methods is used. 
                       
                      FACTORING	
         Unless the factoring is especially difficult, this is the first thing to try. 
                       
                      Here is a quadratic equation that is factorable:             x2 −7x+12=0 
                       
                      First we factor:                                             (      )(     )      
                                                                                    x−3 x−4 =0
                       
                      Next, we use the Zero-Factor Property:                       x −3=0  OR  x−4=0 
                       
                      Solving each equation leads to the solution set:                    
                                                                                   { }
                                                                                    3,4
                       
                      THE	
  QUADRATIC	
  FORMULA        If factoring fails, the Quadratic Formula is used. 
                       
                                                                        −b± b2 −4ac
                                                                   x =         2a           
                                                                               
                      Here is a quadratic equation that is not factorable:         9x2 −6x+5=0 
                       
                      To use the quadratic formula, it is very helpful to first write down what a, b, and c are: 
                       
                                                                            a = 9 
                                                                           b = −6 
                                                                            c = 5 
                                                                               
                      We now evaluate the function using the values: 
                       
                                                                                    2
                                                                    (    )    (    )     ( )( )
                                                             x = − −6 ±        −6 −49 5  
                                                                                ( )
                                                                               2 9
                       
                      We then simplify the formula until we arrive at the solution set.  (Work not shown.) 
                       
                      We can write both solutions separately:   1      2  1    2   
                                                                  {              }
                                                                   3 + 3 i, 3 − 3 i
                                                                               
                      However, the shorthand solution is often acceptable:   1       2   
                                                                               {       }
                                                                                 3 ± 3 i
                       
                                                                                                                                     1 
                       
                       
                      Sometimes a quadratic equation has a missing term.  When this occurs, we don’t have to rely on the 
                      Quadratic Formula.  Although the quadratic formula would still work, there are easier methods. 
                       
                                                                             
                       
                      When solving a quadratic equation with b = 0 the Square Root Property is the easiest method. 
                       
                      THE	
  SQUARE	
  ROOT	
  PROPERTY        If  b = 0 the equation can be written in the form 
                      ax2 +c = 0. 
                       
                      The Square Root Property can be simply stated in this way: 
                       
                                                                       IF  x2 = k , 
                                                                             
                                                           THEN x= k   OR   x=− k . 
                                                                             
                       
                      The following example will illustrate the method:                  2x2 +36 = 0 
                       
                      First we isolate the squared variable:                              x2 = −18 
                       
                      Next we employ the square root property:                            x = ± −18 
                       
                      Often the radical needs simplifying:                                x = ±3i 2 
                       
                      Finally, we write the result in solution set notation:                              
                                                                                         {              }
                                                                                          3i 2,−3i 2
                       
                                                                             
                       
                      When solving a quadratic equation with c = 0 it is always possible to solve by factoring. 
                       
                      FACTORING        If  c = 0 the equation can be written in the form ax2 +bx = 0. 
                       
                      In this situation, we simply factor out an x, then set each factor equal to zero (the Zero-Factor property). 
                       
                      The following example will illustrate the method:                  2x2 +9x=0 
                       
                      Factor out an x:                                                     (       )     
                                                                                          x 2x+9 =0
                       
                      Next, we use the Zero-Factor Property:                              x = 0  OR  2x +9 = 0 
                       
                      Solving each equation leads to the solution set                        9    
                                                                                         {      }
                                                                                          − 2,0
                       
                      Note that zero will always be one of the solutions for quadratic equations in this form. 
                       
                                                                             
                                                                                                                                  2 
         The Quadratic Formula will solve any quadratic equation in standard form.  However, the other 
         methods have survived because they are often easier to use than the Quadratic Formula.  Note that  
         the Completing The Square method is not typically used to actually solve quadratic equations, and  
         not included in this handout.  However, it is still taught due to its applicability in other areas of algebra. 
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