# quadratic equation in $\u2102$

$$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$$ |

for solving the quadratic equation

$a{x}^{2}+bx+c=\mathrm{\hspace{0.33em}0}$ | (1) |

with real coefficients $a$, $b$, $c$ is valid as well for all complex values of these coefficients ($a\ne 0$), when the square root is determined as is presented in the parent entry (http://planetmath.org/TakingSquareRootAlgebraically).

Proof. Multiplying (1) by $4a$ and adding ${b}^{2}$ to both sides gives an equivalent^{} (http://planetmath.org/Equivalent3) equation

$$4{a}^{2}{x}^{2}+4abx+4ac+{b}^{2}={b}^{2}$$ |

or

$${(2ax)}^{2}+2\cdot 2ax\cdot b+{b}^{2}={b}^{2}-4ac$$ |

or furthermore

$${(2ax+b)}^{2}={b}^{2}-4ac.$$ |

Taking square root algebraically yields

$$2ax+b=\pm \sqrt{{b}^{2}-4ac},$$ |

which implies the quadratic formula.

Note. A quadratic formula is meaningful besides $\u2102$ also in other fields with characteristic^{} $\ne 2$ if one can find the needed “square root” (this may require a field extension).

Title | quadratic equation in $\u2102$ |
---|---|

Canonical name | QuadraticEquationInmathbbC |

Date of creation | 2013-03-22 17:36:36 |

Last modified on | 2013-03-22 17:36:36 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 30-00 |

Classification | msc 12D99 |

Synonym | quadratic equation |

Related topic | QuadraticFormula |

Related topic | DerivationOfQuadraticFormula |

Related topic | CardanosDerivationOfTheCubicFormula |