A quadratic equation is a second-order Polynomial

(1) |

(2) |

(3) |

(4) |

(5) |

An alternate form is given by dividing (1) through by :

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

so

(15) | |||

(16) |

Similarly, if , then

(17) |

(18) |

so

(19) | |||

(20) |

Therefore, the Roots are always given by and .

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 17, 1972.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 9, 1987.

Courant, R. and Robbins, H. *What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 91-92, 1996.

King, R. B. *Beyond the Quartic Equation.* Boston, MA: Birkhäuser, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Quadratic and Cubic Equations.'' §5.6 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 178-180, 1992.

Spanier, J. and Oldham, K. B. ``The Quadratic Function and Its Reciprocal.''
Ch. 16 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 123-131, 1987.

© 1996-9

1999-05-25