## The Quadratic Formula

Consider the standard form of a
quadratic equation:

We can solve this general equation by
completing the square
exactly like we would solve it
if the coefficients assumed specific values:

The last of these equations is the celebrated
** quadratic formula**. It can also be written as:

Notice the term

under the square root. It determines the number of solutions of
and their nature, and is called the
**
Discriminant** of .
Clearly there are three cases:

To apply the quadratic formula to a particular quadratic equation just
convert the equation to standard form and substitute the appropriate
values of , and in the formula. However, I believe
it is more efficient and less error prone to make sure you understand
the binomial
formulas and the concept of
completing the square and to solve a quadratic equation from
scratch whenever you need to. Memorizing facts like the quadratic
formula quickly becomes tedious and your memory may fail at an
inopportune moment. On the other hand, there are of course
formulas--like
the distributive law ,
the binomial formulas, and the rules for
manipulating
powers
--that you should be able to remember, but you do so because these
formulas are simple and you use them so frequently that they become
part of your consciousness.