I'd like to answer your questions, but let me answer them in a different order.
-What is a quadratic expression/equation?
ax2 + bx + c = 0
is a typical quadratic expression.
The term "quadratic expression" is a synonym for "second degree polynomial".
I like calling them second degree polynomials.
-What is the quadratic formula/function?
The quadratic formula is a type of trick that is used to find the value of x in the previously mentioned equation. It looks like this:
x = [ -b ± sqrt(b^2 - 4ac) ] / [2a]
Don't worry if the way I typed it doesn't like so nice, it is actually much easier to write by hand than it is to type it.
-What is its purpose and how can it be used in a real life example?
Generally speaking, the purpose of the quadratic formula is to find the value of x for some second degree polynomial.
Specifically speaking, a real life example might involve finding the amount of time it takes for a ball to fall from a tall building, or a car to drive to some location. Often real life examples are physics problems.
-Anything else necessary to be known...
you seem like a good student, keep up the good work.
If you feel like you would like to learn more information than your class covers feel free to contact me.