First of all, let us be
sure we understand what is meant by the term * quadratic
formula*. To do that, we must understand
what is meant by the terms

A * quadratic
expression* is a polynomial of this form:

ax^{2} + bx + c

A * quadratic
equation* is an equation of the form:

0 = ax^{2} + bx + c

Examples of quadratic equations would be:

0*
= *6*x*^{2}*
+ *2*x + ***9**

0*
= *3.2*x*^{2}*
+ *8.2*x *-
1.6

When equations of this
form are solved for * x*, you get
you get the

That is, you can use this calculation
to find the values of * x* that
make the quadratic equation equal to zero. These specific
values of

A calculator for solving the quadratic formula is available here. However, proceed below if you want to know how the above equation that solves for x is derived.

Let us figure out why * x*
is equal to the above value. First, start with the
quadratic formula:

[1]

Subtract * c*
from both sides to get:

[2]

Divide both sides by * a*:

[3]

This next step may look a bit
mysterious at first. It will allow us to turn the left
side of the above equation into a
* perfect
square*. This
method of turning one side of an equation into a perfect
square is often called

Add to both sides:

[4]

* Factor* the
left side of this equation by asking this question:

What two numbers when added are and when multiplied are ?

That number is .

Therefore, we can see the left side of [4] as a perfect square that factors this way:

[5]

The left side can now be written this way:

[6]

Let us do a bit of algebra on the right side of [6]:

[7]

Now, we get a common denominator on the right side:

[8]

Adding on the right side:

[9]

Taking the square root of each side:

[10]

Subtract from each side:

[11]

Do some algebra on the right side:

[12]

Since the terms on the right side now have a common denominator, we will combine those terms by adding them:

[13]

And we are all done. [13] is the solution to the quadratic equation.

A calculator for solving the quadratic formula is available here.