## The Binomial Formulas

The binomial formulas are

and may be any variables, or even more general expressions.
The expression on the left of the first and second binomial formulas
are
** perfect squares**,
the expression on the left of the third
formula is the **difference of **(two)
** squares**.

Note that the first and second binomial formulas are equivalent. Just
replace with to get from one to the other.

It is straightforward to verify the binomial formulas from left to
right using the distributive law. For example:

However, the power
of the binomial formulas arises from being able to read them from right to
left. Some examples will illustrate this. In each case look at the
right expression and think about how to write it either as a perfect
square (as in the first and second binomial formula) or a product of
a sum and difference, as in the third binomial formula. Fill in the
blank space on the left. This page
does not give the answers, however,
click here
if you get stuck or you want to check your answers.

Of the above, equations , , , , and are typical for
the kind of operations that occur in the solution of quadratic
equations.
The equation describes a critical step in the derivation of the
quadratic formula . The other equations described steps that may
occur in factoring

You can find many more examples in any textbook on Intermediate
Algebra. Look for terms like * solving quadratic equations, binomial
formulas, perfect squares, completing the square.*