# The solution of a quadratic equation

The solution of quadratic equation, including complex roots

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Created: 2011-07-23 20:16:59, Last updated: 2021-03-12 16:49:46

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A quadratic equation is a second-order polynomial equation in a single variable x
$ax^2+bx+c=0$
with a != 0

The roots x can be found using the quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

There are three cases:

if $b^2-4ac>0$
then there are two distinct roots, both of which are real numbers

if $b^2-4ac=0$
then there is exactly one distinct real root, sometimes called a double root

if $b^2-4ac<0$
then there are no real roots. Rather, there are two distinct (non-real) complex roots.

The expression
$b^2-4ac$
is called the discriminant of the quadratic equation

Using roots, polynomial equation can be expressed as
$ax^2+bx+c=a(x-x_1)(x-x_2)$

Below the calculator

Digits after the decimal point: 2
x1

x2

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