While searching to learn about complex numbers on the Internet, I was referred also to quadratic equations. Several graphic examples showed how "completing the square" uses a quadratic equation to calculate the length of sides of a square for a new desired area size, which I understood. I also saw examples of how graphing a quadratic equation creates a parabola, which I understood.
How can a quadratic equation represent a square with straight-line sides, AND a parabola with exponential curves? They seem completely different. In fact, I have not found one source, that mentions both in the same article. Is the link or relationship between them, because of the exponential term in the quadratic equation?
To answer your question, let's assume you have a square of side length equal to the value $x$. The area is, as you know, $$Area=x^2$$
Now the above equation tells us that: given a value of $x$, we can always calculate the Area of the square by applying the above formula.
This gives the rise to the fact that "The area of the square is dependent only on the length of its side".
In Mathematics, we call this relationship a function. We could now say that the area of a square is a function of its side length. Let's call the area $A$, and since it depends on $x$, we can say that we have a function that depends on $x$ and looks like this:
$$A(x)=x^2$$
If you specify some x-values you will get corresponding areas for each x, we can write this in the form:
$$(x, A(X))$$
Let's do some:
$$(1,1)$$ $$(2,2*2)$$ $$(3,3*3)$$ $$(4,4*4)$$
If you draw these points (and some more) you will get a curve shaped like a parabola as you earlier said.
The function drawn above showing the function is actually drawn on what is called the Cartesian-Plane which has x-axis and y-axis. It can only be drawn using those axis if we want to use the term function (at least for the sake of this argument).
However, the square shape represents a Geometrical shape that can be drawn on a plane surface without regards to x and y coordinates or the Cartesian Plane at all. It is drawn using lines and angles. This is Euclidean Geometry.
If you draw a square on a Cartesian plane, you will have to use specific line lengths and not a line length of a generic value like $x$. It is a completely different method of representing information, hence the two shapes are different.