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From Whence
        The Quadratic Formula?

                        Don Simmons




© D. T. Simmons, 2009                 1
Deriving the Quadratic Formula
Everybody taking algebra eventually learns the quadratic
formula. But few know where it comes from.

This presentation will help you understand how to get …


               From:    ax 2   bx   c    0


                               b    b2    4ac
               To:      x
                                    2a

© D. T. Simmons, 2009                                      2
Deriving the Quadratic Formula
•    Begin with the quadratic equation…
              ax 2      bx c     0

•    Set the leading coefficient to one.
               ax 2      bx c    0
                           a
                        b    c
               x2         x      0
                        a    a




© D. T. Simmons, 2009                      3
Deriving the Quadratic Formula
•    Isolate the variables on one side of the equation by
     moving the c value.

                   2    b     c
               x          x           0
                        a     a

                        b         c
               x2         x
                        a         a




© D. T. Simmons, 2009                                       4
Deriving the Quadratic Formula
•   Complete the square.
       b      c
    x2   x
       a      a

                             2            2
           b            b        c   b
    x2       x
           a            2a       a   2a




© D. T. Simmons, 2009                         5
Deriving the Quadratic Formula
•    Create a common denominator.
             2  b   b2   b2    4a c
           x      x
                a   4a 2 4a 2  4a a

•    Combine the fractions

                   2    b     b2    b2     4a c
               x          x
                        a     4a2        4a 2




© D. T. Simmons, 2009                             6
Deriving the Quadratic Formula
•    Rewrite in exponential form.

                   2    b         b2      b2      4a c
               x          x
                        a         4a 2          4a 2

                              2
                        b           b2     4a c
                 x
                        2a               4a 2




© D. T. Simmons, 2009                                    7
Deriving the Quadratic Formula
•    Apply the square root property
                              2
                         b              b2         4a c
                   x
                         2a                   4a 2


                    b             b2        4a c
               x
                    2a                 2a




© D. T. Simmons, 2009                                     8
Deriving the Quadratic Formula
•    Isolate x.


               b        b     b      b2        4a c
          x
               2a       2a    2a          2a


                  b      b2   4a c
          x
                         2a




© D. T. Simmons, 2009                                 9
Deriving the Quadratic Formula

• Done!
                                2
                        b   b       4a c
            x
                            2a


© D. T. Simmons, 2009                      10

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Deriving the quadratic formula

  • 1. From Whence The Quadratic Formula? Don Simmons © D. T. Simmons, 2009 1
  • 2. Deriving the Quadratic Formula Everybody taking algebra eventually learns the quadratic formula. But few know where it comes from. This presentation will help you understand how to get … From: ax 2 bx c 0 b b2 4ac To: x 2a © D. T. Simmons, 2009 2
  • 3. Deriving the Quadratic Formula • Begin with the quadratic equation… ax 2 bx c 0 • Set the leading coefficient to one. ax 2 bx c 0 a b c x2 x 0 a a © D. T. Simmons, 2009 3
  • 4. Deriving the Quadratic Formula • Isolate the variables on one side of the equation by moving the c value. 2 b c x x 0 a a b c x2 x a a © D. T. Simmons, 2009 4
  • 5. Deriving the Quadratic Formula • Complete the square. b c x2 x a a 2 2 b b c b x2 x a 2a a 2a © D. T. Simmons, 2009 5
  • 6. Deriving the Quadratic Formula • Create a common denominator. 2 b b2 b2 4a c x x a 4a 2 4a 2 4a a • Combine the fractions 2 b b2 b2 4a c x x a 4a2 4a 2 © D. T. Simmons, 2009 6
  • 7. Deriving the Quadratic Formula • Rewrite in exponential form. 2 b b2 b2 4a c x x a 4a 2 4a 2 2 b b2 4a c x 2a 4a 2 © D. T. Simmons, 2009 7
  • 8. Deriving the Quadratic Formula • Apply the square root property 2 b b2 4a c x 2a 4a 2 b b2 4a c x 2a 2a © D. T. Simmons, 2009 8
  • 9. Deriving the Quadratic Formula • Isolate x. b b b b2 4a c x 2a 2a 2a 2a b b2 4a c x 2a © D. T. Simmons, 2009 9
  • 10. Deriving the Quadratic Formula • Done! 2 b b 4a c x 2a © D. T. Simmons, 2009 10