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

Question

Demonstrate how would you solve a quadratic equation with factoring and provide a quadratic equation for your classmate to solve by factoring

Answer 1

 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:

                    

                   X²  - 23X + 38 =  

                      

                   2 numbers ( m.n) whose sum is 23 and product 38 are ( 19 , 4), therefore:

                 

                   X² - 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 X² - 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.