Introduction to Quadratic Equations.

Definition of a quadratic equation 

A quadratic equation in x is an equation that can be written in the

 form 2 0, , , 0. ax bx c where a b and c are 

real numbers with a + += ≠ 

A quadratic equation in x also called a second degree polynomial equation in x. 

Quadratic formula and its derivation 

Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.^[5] The mathematical proof will now be briefly summarized.^[6] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}$

Taking the square root of both sides, and isolating x, gives:

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

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax² + 2bx + c = 0 or ax² − 2bx + c = 0 ,^[7] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

A lesser known quadratic formula, as used in Muller's method, and which can be found from Vieta's formulas, provides the same roots via the equation:

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

One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case the more common formula has division by zero in both cases.