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```Date: 04/07/2004 at 13:36:10
From: Rachel

As I read various algebra books for high school kids, I find what
appears to be an inconsistent use of the words "quadratic equation",
and I wanted to be sure I use it correctly myself.  Is it correct to
call y = ax^2 + bx + c a "quadratic equation"?  It is an "equation" in
the sense that it sets two expressions equal to each other, however
frequently textbooks seem to call this a "quadratic function" (since
it is a function) and reserve the phrase "quadratic equation" for
ax^2 + bx + c = 0 only.

Are there times when it is more appropriate to refer to y = ax^2 + bx
+ c as an equation, as opposed to a function?  Are the ordered pairs
that fall on the graph of the parabola considered "solutions" to the
equation or to the function?  Can a function have "solutions"?

Is the word "equation" appropriate any time 2 expressions are set
equal to each other?  And is the word "quadratic" appropriate any time
the highest degree of a variable in the equation is 2?  If so, could
the equation of a circle also be considered a "quadratic equation in 2
variables"?  Yet a circle could not be called a quadratic function
since it's not a function.

What about x = ay^2 + by + c on the x-y plane.  This is not a
function.  But can it be called a quadratic equation?

Anything you could say to help clarify this would be much appreciated.

```

```
Date: 04/07/2004 at 16:02:31
From: Doctor Peterson

Hi, Rachel.

You've raised some good questions.  Almost all of them can be answered
consistent with the definitions, but not always with conventions that
kids see, which they can think are more absolute than they really are.

Let's look at the definitions.

An _equation_ is any statement that two expressions are equal.  That
includes

y = ax^2 + bx + c

x = ay^2 + by + c

ax^2 + bx + c = 0

x^2 + y^2 = r^2

Some equations involve only one variable; those can be solved to find
the value(s) of the variable that make them true.  Others involve more
than one variable; we don't usually say that a pair (x,y) for which
the equation is true is a "solution" of the equation, but in some
cases it makes sense to speak that way; we do say we "solve" the
equation for one variable in terms of, or as a function of, the other.

A _function_ is a relation between two variables such that for any
value of the independent variable (in the domain) there is exactly
one value of the dependent variable.  This can be expressed as an
equation like

y = ax^2 + bx + c

which tells how to find y given a value of x; we say we have
expressed y as a function of x.  But this also represents a function:

x = ay^2 + by + c

The only difference is that we have gone against the usual convention
that x is independent; we have instead written x as a function of y.
Note that the equation is not the function; it only represents or
defines it.

When we want to emphasize the function itself, we can name it, and
define for example

f(x) = ax^2 + bx + c

Then we say that the function is f (that is, the relationship named
the distinction:

Are All Functions Equations?
http://mathforum.org/library/drmath/view/53273.html

Also, for the fact that functions do not have solutions (or roots),
while equations do, see:

What Are the Definitions of Zero and Root?
http://mathforum.org/library/drmath/view/64502.html

Finally, something is called _quadratic_ if it involves a second-
degree polynomial. So all the equations I listed are quadratic; some
are univariate (one variable), others bivariate (two variables).  At
high school level, texts (or students) tend to equate "quadratic
equation" with the univariate kind, and in particular with one in
standard form, since that is what they spend inordinate amounts of
time learning to solve.  For an example of bivariate quadratics,
a.k.a. conics, see

Compare that with

In summary, there are different kinds of "quadratic equations"; as
with most definitions in English or in math, the term depends somewhat
on the context.  When the book is talking about solving univariate
equations, they can say "a quadratic equation has the form ax^2 + bx
+ c = 0" as if that were the only way the phrase is used, because for
the moment it is; but later when they want to graph quadratic
functions or quadratic curves, they can use the word a little
differently.  So you should try to be consistent within your context,
and using words according to their definitions; but don't worry too
much about being completely uniform across the board.

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

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

```

```
Date: 04/07/2004 at 17:02:02
From: Rachel
function)

Thank you Dr. Peterson!  I appreciate the thorough response, the quick
turn-around, and the fact that you are there to answer these
questions.  You have cleared things up for me.  I can now use these
terms and feel confident I am using them correctly.
```
Associated Topics:
High School Basic Algebra
High School Definitions

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