PlanetMath: quadratic equation in $\mathbb{C}$

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quadratic equation in $\mathbb{C}$

(Theorem)

The quadratic formula

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

for solving the quadratic equation

$\displaystyle ax^2+bx+c = 0$

(1)

with real coefficients

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 and adding to both sides gives an equivalent equation

$\displaystyle 4a^2x^2+4abx+4ac+b^2 = b^2$

$\displaystyle (2ax)^2+2\cdot2ax\cdot{b}+b^2 = b^2-4ac$

or furthermore

$\displaystyle (2ax+b)^2 = b^2-4ac.$

Taking square root algebraically yields

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

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).

"quadratic equation in $\mathbb{C}$ " is owned by pahio.

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Other names:

quadratic equation

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Attachments:: biquadratic equation (Topic) by pahio

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Cross-references: field extension, characteristic, fields, implies, taking square root algebraically, equation, sides, square root, complex, coefficients, real, quadratic formula
There are 19 references to this entry.

This is version 7 of quadratic equation in $\mathbb{C}$ , born on 2007-11-01, modified 2007-11-05.
Object id is 10026, canonical name is QuadraticEquationInMathbbC.
Accessed 1062 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	30-00 (Functions of a complex variable :: General reference works )

Pending Errata and Addenda

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