What is a quadratic equation and what is its simplified and cannonic form?

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See here. – David Mitra Aug 8 '12 at 23:56
I didnt get which one was tha canonic form and which one was the simplified – user1038739 Aug 9 '12 at 0:01

A quadratic equation (in one variable) is a polynomial equation $P(x)=0$, where $P(x)$ is a polynomial of degree 2.

The canonical form of a quadratic equation is $ax^2+bx+c=0$. The simplified form I'm not sure about, but I'm guessing $x^2+bx+c=0$ (obtained from canonical by dividing both sides by the coefficient of $x^2$).

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@user1038739 terminology: the simplified polynomial $x^2 + \dfrac{b}{a} x + \dfrac{c}{a}$ is also known as the monic associate. – user2468 Aug 9 '12 at 0:14
Maybe "simplified" is in the eye of the beholder, but I consider the form where the square is "completed" to be simple. That is, $$a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}$$ – J. M. Aug 9 '12 at 1:08

Quadratics are usually written in the following three ways:

Expanded form

This form is written canonically as

$$f(x)=ax^2+bx+c$$

This is a very general form and it is easy to find $f(0)$ if needed. However, finding the roots are a bit more difficult.

Factored form

This form is written as

$$f(x)=a(x-x_1)(x-x_2)$$

This form is very convenient if you need the roots of $f(x)$, as they are simply $x_1$ and $x_2$.

Vertex Form (complete the square)

This form is written as

$$f(x)=a(x-h)^2+k$$

This form is convenient for finding the vertex $(h, k)$. This is the form that is used to derive the quadratic formula.

Expanded form is usually considered to be the most basic form, so this is probably what you want.

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