Definition of quadratic equation? What is a quadratic equation and what is its simplified and cannonic form?
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.
In Algebra, Equation with the highest degree 2 is a quadratic equation. And its simplest form is 
$$
ax^2+bx+cx = 0,
$$ 
where $x$ represents variable and $a$,$b$, and $c$ represents constant or numbers. Also, $a$ should not be equals to zero because if $a = 0$ then the equation will become a linear one. And its graphical representation is a parabolic shape.