Definition of quadratic equation?

Question

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

Answer 1

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 $x^2$).

Answer 2

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.