Quadratic equation vs quadratic function

[Asam]

Hello,
I have difficulty to understand the difference between the meaning of a quadratic equation and the meaning of quadratic function. I'd like to describe my question with the following example.

Assume that there is a football field with area of 700 sq. m. And the length of it is 30 meters longer than the width. then we will have;
X(X+30)=700

X2+30X-700=0

This will be our equation. To convert it to a function, I assume we replace “0” with “y”

Y= X2+30X-700

My question is;

In the equation we talk about an area, but how it is possible with a slight configuration we give it a totally new meaning. Where recognize a relationship between 2 variables.

Please, enlighten me.

 

[Dr.Peterson]

Hello,
I have difficulty to understand the difference between the meaning of a quadratic equation and the meaning of quadratic function. I'd like to describe my question with the following example.

Assume that there is a football field with area of 700 sq. m. And the length of it is 30 meters longer than the width. then we will have;
X(X+30)=700

X2+30X-700=0

This will be our equation. To convert it to a function, I assume we replace “0” with “y”

Y= X2+30X-700

My question is;

In the equation we talk about an area, but how it is possible with a slight configuration we give it a totally new meaning. Where recognize a relationship between 2 variables.

Please, enlighten me.

If I wanted to express this problem in terms of a function, rather than an equation in one variable, it would be

A = x(x+30)​

Solve A = 700​

Here we are expressing the area as a function of width, and then finding when that area is 700.

In your version, y represents the excess area over 700.

Incidentally, you can write "x squared" as "x^2".

 

[Cubist]

Here are the "definitions" that I keep in my own (weird) brain...

Equation

An equation simply shows equality. The algebraic expressions on the left and right hand side of the = symbol yield the same value. These values don't have to make any kind of "intuitive sense" in the context of any particular problem.

Examples...

Z=pi + 10 this equation fixes Z's value
Z - pi=10 this is a different equation to the above. However, it's a manipulation of it and therefore it still implies the same thing (that the value of Z is pi+10)

Y = X² + 30X - 700 this equation links the values of two variables X and Y. As one variable changes the other one changes. Quite often we'd also call this a function because it's in the form of <a variable> = <an expression in terms of a different variable>

Function

Here's a function that returns the area of a field that is 30 meters longer than its width
f(X) = X(X + 30)

Here's a function that returns the area of the same field minus 700m²
g(X) = X(X + 30) - 700

The second function is not as useful in a "general" sense as the first one. Therefore, like in @Dr.Peterson post#2, we often choose to declare a function in a way that makes it most useful (and in most situations). But, ultimately, a function can return any value that you desire and ideally you'd state the purpose (and inputs/ outputs) of the function in words somewhere alongside it.

When graphing or manipulating a function then (often) we'd let y=f(x)

 

[JeffM]

It seems to me that there is little to enlighten you about. You are almost there.

An equation is a statement that two mathematical objects are the same.

A function describes a relationship between one mathematical object and one or more other mathematical objects. It is frequently expressed as an equation.

The difference you are looking for is the difference between a variable and an unknown. The function described by the equation

[math]y = 3x^2 - 3x + 1[/math]
can be EXEMPLIFIED, but has no solution. It expresses a relation between two numbers, one called x and one called y. Any pair of numbers that have the numeric relation specified is an example of that function. Thus, if we change x we may change y. And so calling x and y “variables” makes sense.

Now the equation

[math]3x^2 - 3x + 1 = 19[/math]
has a solution, which means that there exist a limited number of numbers that make the equation true.

A function usually summarizes through an equation a relationship that is general.

A problem poses through an equation a question that, we hope, has a quite limited number of answers