If the coefficients are arbitrary complex numbers, then this is all that we can say:

Discriminant: | Solutions: |
---|---|

Zero, | One solution; |

Nonzero, | Two solutions. |

If the coefficients are all real numbers, then we can say more:

Discriminant: | Solutions: |
---|---|

Zero, | One real solution; |

Positive, | Two real solutions; |

Negative, | Two conjugate imaginary solutions. |

If the coefficients are all rational numbers, then we can say even more:

Discriminant: | Solutions: |
---|---|

Zero, | One rational solution; |

Positive perfect square, | Two rational solutions; |

Positive non-square, | Two conjugate irrational real solutions; |

Negative, | Two conjugate imaginary solutions. |

If *a* = 1 and the coefficients are all integers,
then we can say yet more:

Discriminant: | Solutions: |
---|---|

Zero, | One integer solution; |

Positive perfect square, | Two integer solutions; |

Positive non-square, | Two conjugate irrational real solutions; |

Negative, | Two conjugate imaginary solutions. |

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