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[parent] 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 $ 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

$\displaystyle 4a^2x^2+4abx+4ac+b^2 = b^2$
or
$\displaystyle (2ax)^2+2\cdot2ax\cdot{b}+b^2 = b^2-4ac$
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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See Also: quadratic formula, derivation of quadratic formula, Cardano's derivation of the cubic formula

Other names:  quadratic equation

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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 MSC12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 30-00 (Functions of a complex variable :: General reference works )

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