What is a Quadratic Equation?

Date: 08/04/97 at 14:38:14
From: Robert M. Thacker
Subject: What is a quadratic equation anyway?

Dear Dr. Math:

In plain English, can you explain to my daughter, "What is a quadratic 
equation?" What is it used for? How can we use it to solve everyday 
problems?  

Please do not use a lot of mathematical gibberish in your explanation.  
Thank you.  

Robert M. Thacker

Date: 08/05/97 at 15:13:48
From: Doctor Ceeks
Subject: Re: What is a quadratic equation anyway?

Hi,

Before I answer, could you please indicate something about the
level of your daughter's mathematical knowledge?  Is she in
Algebra I, or is she in kindergarten?  Does she like math or is
she afraid of it?

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 08/05/97 at 15:33:20
From: Robert M. Thacker
Subject: Re: What is a quadratic equation anyway?

Dear Dr. Math:

She will begin Algebra I next year, and likes math in general.  
The problem is I cannot give her a simple, practical, commonsense, 
applied answer to what a quadratic is.  

Sincerely,  

R.M. Thacker

Date: 08/05/97 at 16:36:43
From: Doctor Ceeks
Subject: Re: What is a quadratic equation anyway?

Hi,

It's difficult to know how to approach this question.

Perhaps if I give two examples of where a quadratic equation would
arise it would help.

Here are two examples:

1.  I'm thinking of two numbers.  Their sum is 10 and their product is
    21.  What are the two numbers?

    More generally, two numbers have sum S and product P.  What are
    the two numbers?

To solve this, you will end up naturally with a quadratic equation:

    x^2 - 10x + 21 = 0  or  x^2 - Sx + P = 0, respectively.

There will generally be two solutions to a quadratic equation
corresponding to the two numbers.  

(In fact, all quadratic equations essentially arise in this manner.  
I say essentially, because generally, people consider the equation

    Ax^2 + Bx + C = 0

to be the most general quadratic equation, where A, B, and C are
constants with A not equal to zero.  However, if you divide this
throughout by A, you will get an equation looking like x^2-Sx+P = 0
if you take S = -B/A and P = C/A.)

2.  A square picture frame contains a picture with a mat border.
    The border is 3 inches thick on the sides and 4 inches thick
    on the top and bottom. If the area exposed within the mat border
    is 528 square inches, what are the dimensions of the original  
    frame?

Again, a quadratic equation will arise naturally.

(In this case, x^2-14x-480 = 0.)

Other places where a quadratic equation may surface come from
geometry, such as trying to find the intersection of a line and
a circle (an example of this arises in one of the ways people find
all the Pythagorean triples).  Also, quadratic equations sometimes
occur in physics when studying how objects fall to Earth.

If your daughter knows what a graph is, you can toss a ball and
note that the ball follows a path which looks like the graph of
a quadratic function.  If you have access to a blackboard, you can
draw, as accurately as you can, a graph of the quadratic function
f(x) = -x^2/36.  Draw it so that you get a nice section of it on 
the board. Then take a ball (or any object) and, with practice, you 
can toss it accross the blackboard so that it exactly follows the 
graph you have drawn.  It's pretty exciting to see this done.

(Note that a quadratic _equation_ is an equation you get when you
set a quadratic _function_ equal to zero.)

Being able to solve quadratic equations was one of the early 
challenges to the human race and was solved by the ancients... so 
useful was their discovery that the quadratic formula (which gives 
the solutions to a quadratic equation) has been remembered and 
handed down over the centuries.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 12/10/2002 at 11:22:57
From: Joey Greer
Subject: Arriving at the quadratic equation naturally

Hi, 

Above it says:

I'm thinking of two numbers.  Their sum is 10 and their product is 21.  
What are the two numbers?

More generally, two numbers have sum S and product P. What are the two 
numbers?

To solve this, you will end up naturally with a quadratic equation:

    x^2 - 10x + 21 = 0  or  x^2 - Sx + P = 0, respectively.

I started thinking about this and wondered how the ancient 
mathematician deriving that equation arrived at that naturally. How 
would he know that the equation would have two answers and the two 
answers combined using the given operation (multiplication or 
addition) would be the answer to the question? Did he come to it 
somehow from another equation? Because I know that if I didn't know 
the quadratic equation is used to solve questions like that and I were 
asked it, I would not have derived that equation. It seems like a 
phenomenon that it actually even works out as it does. Can you please 
help me to understand this?  

Sincerely,
Joey

Date: 12/10/2002 at 12:30:20
From: Doctor Peterson
Subject: Re: Arriving at the quadratic equation naturally

Hi, Joey.

Here is how I would get from the problem to the equation. It is not 
just a leap based on knowing what kind of equation to expect, but as 
Dr. Ceeks said, it arises naturally.

  I'm thinking of two numbers.  Their sum is 10 and their product
  is 21. What are the two numbers?

Suppose we call one of the numbers x. Then, since their sum is 10, 
the other number has to be 10-x. The fact that their product is 21 
tells us that

    x(10-x) = 21

Simplifying this, we get

    10x - x^2 = 21

    x^2 - 10x + 21 = 0

There's the equation.

Now, the work would have been a little different for the first people 
who used such equations, since they didn't have algebraic notation, 
but the underlying ideas would be the same.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/