The Quadratic Formula

Here are a few vids on the Quadratic Formula itself. How to derive it and why it looks like it does. There will be a couple of more short vids on this topic, but for now here are these.

What are we doing? Figuring out what x is when I have a 2nd degree polynomial in standard form.

There needs to be several videos and one or two after these three. If you haven't already look over the page on completing the square. It too needs work, but it has a video that shows you how to complete the square and brings us ever closer to the quadratic.

The first vid shows the concept of taking half of "b" and completing the square...the quadratic formula is just advanced completing the square. And it's beautiful because it takes ANY quadratic equation and tells us what x is just by knowing how many x², how many x and how many units we have.

For example:

a = 1, b = 3, and c = 2. a is the number of x², b is how many x and c is the number of units. I suggest you use the quadratic formula on easy one like that first. But before we get ahead of ourselves here are two little kids getting the basic concepts down. The idea being if little kids can "get it" so can older kids and adults. Note here on this page I am just using the symbols with no pictures...in the vids you get the pictures. You can see

is a perfect square.

In this example:

a = 2, b = 5, and c = 3. and then you just stick the numbers into the quadratic formula (shown below) and do the math. You may note many just memorize it without knowing where it came from or what it means. I've actually had arguments with college professors (teachers at a community college) about what it means to complete the square. They were also blissfully unaware that completing the square and the quadratic were integrally related.

Anyhow here is the formula:

Start with ones that are easily factorable so the students get a feel for the formula and they can see if they got it right.

This second vid really needs redoing because you can't always see the symbols and I left off the x in the middle term when I copied it at one point...but then I'd have to redo the third video (on this page, second video in that set of 3) too, so good enough for now. What you see is dividing by "a" so no matter what we come out with just one x². Also I make up for it in the next video because I recap this vid:

The second EASIER way is to multiply to make sure you get a square amount of x²'s (the "a") and an even number of x. (the "b") throw c (the units) over there with zero and solve for x:

The third and "hardest" or longest way comes from pre-calculus and standard form for the vertex:

Applying The Quadratic Formula

I can hear you already. So here are a few shots and a short screencast going over the problems one by one. Four examples for you...

The first two allow the student to check their work for themselves.

Watch the screencast just for good measure if you have questions about the pictures because toward the end I skip some steps, just like you will as you become more and more proficient with the quadratic formula. But to start all the steps are included and the problems are easy!

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