The Quadratic Formula and Imaginary Numbers

In the past two days of class, we touched on two major topics:

-How to apply the Quadratic Formula to quadratic equations, along with the parabolas that accompany them (along with the memorization of the formula);
-Imaginary numbers, what they are, what they mean and how to apply them in mathematics.

So, to dive right in--

The Quadratic Formula
As I'm sure you have heard before, the Quadratic Formula is a formula that allows you to plug in values "a", "b", and "c" from an equation, solve it, and from there obtain your x values.

Here it is:

This equation is used to find the value of x in a quadratic example.

a=coefficient of x^2
b=coefficient of middle term x
c=third term

You simply plug these values in for a particular equation, solve it, and out comes your (usually) two x values.

An Example:

The first step of this is to determine the value of y. Say y is zero, and the rest is easy. Here are the values for a, b, and c:

a=1

b=-10

c=9

Then, you plug these values into the quadratic equation (above). This comes out to:

x=(10 +/- sqrt[100-36])/2

x= (10+/- 8)/2

x= 9, 1

I hope this quick overview of the Quadratic Formula has helped in solidifying your understanding. 

Quadratics in Parabolas

Every quadratic equation, when graphed, comes out to a parabola; meaning that it resembles some kind of a "half-pipe" in its line. There are three kinds of parabolas in quadratics (disregarding sideways vs vertical), displayed below:

Let's take a look at the graph of the quadratic solved previously in the post;

As you can see, the quadratic x^2-10x+9 is a parabola as it resembles a half-pipe. If you take a look at where the line intersects the x-axis, you'll see that those points are the two solutions of the equation. This applies to all quadratics looking for an x value.

The number of solutions a quadratic has (one, two, or zero) is based on the discriminant in the Quadratic Formula. This is the b^2-4ac, of course with the a, b, and c values plugged in the with the relative equation.

Here is a chart developed in class that distinguishes between one, two, or zero solutions in a quadratic:

Note that the "one x-intercept (solution)" side includes two intercepts as well. No x-intercepts include negative square roots.

Imaginary Numbers
Contrary to its name (and the several puns that accompany it; I'll spare them here, for the most part) imaginary numbers are very much real.

They are represented by i; an abbreviation for a definition that is actually very simple:

i=sqrt(-1).

Yup, that's it.

Here is a list that we brainstormed in class about imaginary numbers and what they mean (after a quick internet source on our fancy imaginary--I mean i--Pads).

Here is a diagram with the explanation of what "complex numbers" really are (we didn't seem to understand it at first):

And some quick work with i and its exponents, following a remarkable pattern:
i=i
i^2=-1
i^3=-i
i^4=1
i^5=i

I hope this post has helped your understanding of Quadratics, Parabolas, and our newly-discovered topic of imaginary numbers.

--Peter