A quadratic expression has three common presentations.

- Standard (or polynomial) form:
- Factored form
- Completed square (or vertex) form

Let’s see the expression in these three forms:

- Standard form is simply
- Factored form
- Completed square form

These three forms are equivalent. No matter what value of you might choose, the three forms will compute to the same value.

Let’s check with

Now, spot checking like this is not a proof. I could have engineered that 10 would work, yet nothing else does. Beginning with one form, and algebraically changing its form will constitute a proof. We’ll do that work at the end of this page.

Let’s see the graph of this function, .

Notice that the intercept is the constant in the standard form .

Notice that the intercepts seem to be related to the factored form, .

Notice that the vertex (turning point) seems to be related to the completed square form .

In this unit, we find out how to algebraically convert from one form to another, and find out what each form is particularly useful for.

# Proof of Equivalence

Show that

Show that

As Euclid postulated a long time ago, things that are equal are equal to each other. So if and then

Therefore all three expressions are equivalent.