Everything Quadratic

Question

In my 8th grade Advanced Algebra class, my group is moving extremely slowly due to constant chattering. I find it frustrating and would like to have the math skills to go into Advanced Geometry next year in high school. My teacher has often mentioned how we will learn about the quadratic formula this year, but it is not very likely, as we only have a month left in school.

I would like to learn:

-What is the quadratic formula/function?

-What is its purpose and how can it be used in a real life example?

-What is a quadratic expression/equation?

-Anything else necessary to be known...

Thank you.

-Yasmine

Answer 1

quadratic formula is theoretically important but has uses (although computers tend to be used nowadays for simplicity).

 

Physics comes to mind: besides the fact that parabolas model kinematic motion (trajectory of falling objects), quadratic functions appear in subjects in which it may not be intuitively clear at first sight that quadratics apply:

 

for instance, any second-order linear differential equation with constant coefficients can be solved via quadratic formula (these "ODE"s as they are commonly known as model the physics of an electric circuit, or oscillating spring, and so on)

Answer 2

Hi Yasmine,

 

I'd like to answer your questions, but let me answer them in a different order.

 

-What is a quadratic expression/equation?

Generally speaking,

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. 

 

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I hope you found this information helpful!
-Michael
http://www.wyzant.com/Tutors/Michaels_Tutoring

 

Answer 3

Hello Yasmine, I will try to answer your questions though not necessarily in order.

 

A quadratic expression is a polynomial whose degree is 2.  You are used to linear equations I'm sure, where the degree is 1.  The degree of a polynomial is the highest exponent that appears in it.

So the simplest quadratic expression is simply x^2.  You can add polynomials of lower degree and the result is still a quadratic expression.  So you can have x^2+ x + 1.  (Constants are considered polynomials of degree 0 since x^0 =1 for every x).  You can also multiply any term by a constant so the most general form of the quadratic expression is:

ax^2 + bx + c, where a, b and c, are constants.

A quadratic function is simply a function of that form: f(x) = ax^2+bc+c

Anytime you have that, it is a quadratic function.

 

The quadratic formula is a formula that "solves" the quadratic equation.  What does it mean to solve an equation?  In mathematics the "solutions" of a a function are the values of x that make that function = 0.  You will hear of solutions and zeroes of a function used this way if you haven't already.  For a linear equation, f(x) = mx +b, the solution is simple:

0=mx+b means

-b=mx and so

x= -b/m.  Note that this is the value of x when f(x) = 0

For the quadratic formula you need a little more to solve it.  You have to know a trick called "completing the square," which I am sure you will have to do examples of with actual quadratic equations before the formula is introduced.   I am attaching a pdf I made some time ago outlining how to derive the quadratic formula.

That is it shows how to go from 
0=ax^2+bx+c, to two values of x that make the equation 0.  Note that the values are not always real numbers(they can be complex, eg 2+3i).

 

 Edit: on second thought I do not see a way to attach a pdf.   I will upload it to my resources in case you want to see it. 

Here is a link to the file:
http://www.wyzant.com/resources/files/275058/deriving_the_quadratic_formula