Sums And Products of Roots

While tutoring a friend on sums and products of roots of equations, I came upon an interesting result.  It began with a few formulas to help with common questions such as:

      1 / α + 1 / β = (α + β) / αβ = (-b / a) / (c / a)

                                                = -b / c

      α² + β² = (α + β)² - 2αβ

                   = (-b / a)² - 2c / a

                   = (b² - 2ac) / a²

      and finally:

      αβ² + α²β = αβ(α + β)        = c / a * (-b / a)

                                                  = -bc / a²

From this I started manipulating a few others until I came to this:

                                 (α - β)² = α² - 2αβ + β²

                                              = (α² + β²) - 2αβ

                                              = (b² - 2ac) / a² - 2c / a

                                              = (b² - 4ac) / a²

                                     α - β = ±√((b² - 4ac) / a²)

                                              = (±√(b² - 4ac) / a

                      α - β + (α + β) = (±√(b² - 4ac) / a) + (-b / a)

                                         2α = (-b ±√(b² - 4ac)) / a

                                           α = (-b ±√(b² - 4ac)) / 2a

Look familiar?

Yep, the quadratic formula!  Too easy.