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When solving a quadratic equation by factoring, what property do you need to use.

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 Given  :
                 aX2 + bx + c 
 
                 to choose 2 numbers such as m. in such a way that
 
                   m+ n = - b/a
                   m. n = c/a
 
                 If this equation has solution with integer values, then equation can be factored to
 
                   a X2 + bX + c = a( x -m )( x-n)
 
                    Example: a =1:
                    
                   X - 23X + 38 =  
                      
                   2 numbers ( m.n) whose sum is 23 and product 38 are ( 19 , 4), therefore:
                 
                   X- 23X + 76 = (X -19) ( X-4)
 
                     In order for a quadratic to be factorable to binomial with integer values.
 
                        It is necessary and sufficient that -b to be Sum of the products of 2 factors of C:
 
                          In here we write the number 38 as product of its prime factors as:
 
                            76= 2 . 2 . 19   factors are (2, 4, 19)
 
                             we see that 76 = 4 *19    4+ 19 =23 , therefore the quadratic is factorable as above.
 
                     Now consider the following:
 
                   3 X2 - 14 X + 15 =
 
                    Now we have to have 2 numbers is 14, and product is 45.
 
                     The numbers by calculating in our mind are (5,9)
 
                     We break up the 14 into 5+9
 
                   3X2  - 9X -5X +15 =
                   
                  We factor by part:
                 
                   3X( X -3 ) - 5 ( X - 3) =
                    Now Common factor is ( X-3) , therefor
 
                   3 X- 14X + 15 = ( X- 3) ( 3X - 5 ) 
 
                      If :
                               X2  + 5X - 7 
                              Here there is no 2 integers where their Sum is +5, and product is -7, in this case we
                             factor by what is called : Factoring by Completing square which gives us factors of
                               irrational and complex numbers.