## Quadratic Equations

A quadratic equation can be written in the ** standard form**

where is the unknown variable, the
** coefficients** , and are real numbers, and
(If we have a
linear equation .
and a much simpler problem.)
In a class like this the coefficients are usually given explicitly,
but in actual applications they are often variables, or even
algebraic expressions.
To **solve** the quadratic equation means to find values of that
make the equation true.
To illustrate the principles and issues let's look at a special case
first. Consider the equation

You might say that's not a quadratic equation since it's not in the
form .
However, it can be **converted** into an ** equivalent**
equation that is in that form, by
suitable operations on both sides of the equation:

The last equation in this sequence **is** in standard form, with ,
, and having the given values. However, the equation
can be solved much more easily than
:

Thus there are two solutions of the equation, and .
We can (and should) verify this by substituting these values in the
original equation. If we obtain and if
we obtain
.

Note the symbol in the second and third of the above sequence
of equations. The square root of is positive by convention and
equals . However, our task at that stage is not to compute a
square root as such, but to answer the question for what values of
does equal ? There are two such values,
and , and we must consider both possibilities.

Let's now consider the more general equation

where and are considered known and, as before, needs
to be determined.
It can be solved just like the special case considered earlier:

** The key to solving quadratic equations is to convert them to
the form . This process is called **** completing the
square.** It is based on the first and second
** binomial formulas .**

Let's see how this works with our equation in standard form:

If the constant term was instead of we would have a
perfect square . To make it so we just add on both sides and obtain

which can be rewritten as
Note that since

we simply look at the factor
of , halve it, square it, and add the appropriate constant that
makes the constant equal to that desired value. One simple minded way
to obtain that constant is to subtract whatever constant is there, and
then add the desired value. For this to work the leading coefficient
(multiplying ) must equal . If it doesn't we divide first
by the leading coefficient on both sides. (However, do not memorize
this procedure as a recipe. Rather think about it, and practice it,
until it makes sense and is so compelling that it becomes a natural
part of your repertoire.)

### Two Real Solutions

An example illustrates the process:

We easily check that and do in fact satisfy the
original equation.

The above example illustrates one of three possible outcomes of this
procedure, the case where there are two real solutions.

### One Real Solution

In the following example there is only one solution:

The reason there is only one solution is the fact that there is one
and only one number whose square is .

### Two Conjugate Complex Solutions

If the above procedure leads to taking the square root of a negative
number we obtain a conjugate complex pair of solutions as illustrated
in the following example:

** The Quadratic Formula**

Of course there is nothing to stop us from applying this procedure
to the
general
equation . This gives rise to
the quadratic formula .
Personally, I prefer not to burden my mind with having to memorize
reliably yet another formula, and so I complete the square almost
every
time I solve a quadratic equation.