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# quadratic equation in $\mathbb{C}$

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

for solving the quadratic equation

$\displaystyle ax^{2}\!+\!bx\!+\!c\;=\;0$ | (1) |

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

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

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

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 similar quadratic formula is meaningful besides $\mathbb{C}$ also in other fields with characteristic $\neq 2$ if one can find the needed “square root” (this may require a field extension).

Related:

QuadraticFormula, DerivationOfQuadraticFormula, CardanosDerivationOfTheCubicFormula

Synonym:

quadratic equation

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

30-00*no label found*12D99

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