Quadratic Equations 
                                                              
                                                            
                     I.     Quadratic Equations 
                            a.  Definition:  A quadratic equation with one unknown variable is an 
                               equation in which there appears an exponent of 2 on the unknown 
                               (and sometimes an exponent of 1 as well). 
                                       For instance:      x2 40    is quadratic 
                                                           x2 2x0  is quadratic 
                                                           x2 2x10  is quadratic 
                                                           x2 14x2 2x   is quadratic 
                                                            
                            b.  The General Form of a quadratic equation is:ax2 bxc0. 
                                       acoefficient of x2 term 
                                       bcoefficient of x term 
                                       c constant term (a number) 
                                      
                      
                     II.    Methods of Solving Quadratic Equations: 
                            a.  Put equation in standard form. This may involve removing 
                               parentheses, combining like terms, and moving all terms to one 
                               side of the equation. 
                            b.  If the left-hand side factors, set each factor equal to zero and solve 
                               the 2 linear equations. Then check your answers!! 
                                
                                     x2 3x40
                               Ex)    
                                     (x  4)(x 1)  0
                                     x40            x10
                   
                                                 or 
                                     x  4           x 1
                                
                                     Answer:  x4, x1 
                                
                   
                               Ex)         2
                                     (x2) 4
                                     x2 4x44
                                     x2 4x 0                  Equation is now in 
                                                                standard form. 
                                       
                                     x(x4)0
                                                 
                                     x  0  x  4
                                     Answer:  x0, x4 
                   
                   
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                      c.  If the left-hand side does not factor, use the quadratic formula to 
                         solve the equation. Then check your answers!! 
               
                          Quadratic formula: 
                                        b b24ac
                                    x 
                                              2a
               
               
                         Ex)    x2 3x10  (This will not factor!) 
                             a1    b3  c1 
                              
                                3 324(1)(1)
                             x      2(1)     
                          
                             x  3 94  
                                   2
                          
                             x  3 5 
                                  2
                          
                             There are two answers for x:  x  3 5   or  x  3 5  
                                                          2            2
               
               
                         NOTE:    If the number which appears under the radical in the 
                                  quadratic formula is negative there is no solution for x, 
                                  since it is impossible to take the square root of a 
                                  negative number. 
                          
                          
                          
                          
                          
                          
                          
                          
                          
                          
                          
                          
                          
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                        III.   More Examples: 
                         
                               1.   x2 3x0                              This factors. 
                                    x(x3)0 
                                    x 0    or     x30 
                                                 x 3      
                                   Answer:  x0, x3 
                                
                               2.  x2 90                                This factors. 
                                    (x3)(x3)0 
                                    x30    or     x30 
                                    x3             x 3 
                                   Answer:  x3, x3 
                                
                               3.  x2 12x360                           This factors. 
                                    (x6)(x6)0 
                                    x 6    or     x  6     (both answers are the same) 
                                   Answer:  x6 
                                
                                                                    This factors and may be solved by factoring but 
                               4.  3x2 x20 
                                                                    you can solve any quadratic by the quadratic 
                                                                    formula. It will be illustrated with this example 
                                   Method 1: Factoring                     
                                    (3x2)(x1)0                   that you will obtain the same answers from either 
                                                                    the quadratic formula or by factoring. 
                                   3x2           x 1 
                                    x  2           
                                         3
                                
                                   Method 2: Quadratic Formula 
                                   3x2 x20 
                                    a3   b1  c2 
                                        (1) (1)2 4(3)(2)
                                    x             2(3)             
                                    
                                    x 1 124 1 25  
                                            6            6
                                    x 15   or      x  15   
                                         6                6
                                
                                    x 1    or       x  4  2  
                                                       6     3
                                    
                     
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                  Practice Problems: 
                  Solve the following quadratic equations by factoring. 
                  1.  x2 2x 0                                  
                   
                  2.  p2 90                                    
                   
                  3.  x2 6x80                                 
                   
                  4. T2 6T 160                                
                   
                  5. q2 5q 6                                  
                   
                  6. 4y2 25 
                   
                  7. 9d2 6448d  
                   
                  8. 3e2 3e360 
                   
                  9. 30h2 23h4 
                   
                  10. (5a1)2 9 
                   
                  11. (5s15)2 64 
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