Quadratic Factoring and More

Through these classes, there has been a collection of vocab words:
Factoring Quadratics: finds the roots  of a quadratic equation. Factoring quadratic equations in standard form, , can often be accomplished by finding two numbers that add to give b, and multiply to give ac
Factoring by Grouping: means that you will group terms with common factors before factoring as well as common factors inside of a parenthesis. 

Over the past few classes, D block covered and reviewed a couple key topics. First we looked another way or secret to factor an expression. By adding and subtracting a specific factor (which is only adding zero to the expression) you can absorb one of the added factors and continue to solve and factor.

Example #1:

X^4+X^2+1 hint: add and subtract X^2

X^4+X^2+1+X^2-X^2

Now you have to absorb one of the factors that you just added to the expression.

X^4+2X^2+1-X^2

(X^2+1)^2-X^2

Now this is just difference of two squares

A=X^2+1

B=X

=(X^2+1+X)(X^2+1-X)

Example #2:

X^4+4 hint: add and subtract 4X^2

X^4+4+4X^2-4X^2

A=X^2

B=4

(X^2+4)^2-4X^2

(X^2+4-2X)(X^2+4+2X)

Another topic was introduced to the class called Factoring Quadratics.

Here is a video on how to factor a quadratic a basic but helpful way

Click Here to watch video 

There are a couple of ways that you can speed the process up and the way that was taught in class is in a table form. But first here are three example that I think give a good range as to how to factor quadratics with using the table method to help.


Example #1: This is a simple and great star as were get into factoring quadratics
X^2+8X+12
In this case a= (X^2), b=(8X), and c=(12) but we are only going to use the actual consistence of each term listed and bring the variables back later.
These terms finally become a=1, b=8, and c=12
Now is how you use those terms:
Make a table like the one below, substitute the letters (a,b and c) for the terms that equal them, and multiply as shown.
a*c=12
b=8

As you can see, for the first column, I looked for all the factors of 12 and saw if any could be added up to 8. 
So now here comes the math part :)
(X+6)(X+2)
And if you are unsure about your solution, you can FOIL out the problem

Example #2: This is a great one that gets a little more challenging from switching up the negitives and positives, but also just makes sure that you are doing the steps correctly
X^2-12X+20

Now, if you look at the chart that I made, you can look for the once that match up. And in this case the factors will be 2*10. But, as you can see there can be different answers when adding or subtracting these numbers. If you look at the original equation, you can see that (+20) and that means that both of the signs will be the same so the numbers will be -2*-10
(X-2)(X-10)

Example #3: As you can see, there is a coefficient before the first term, this does complicate things just a bit but also can give you reassurance on what you need to work on and what you are good at with these steps.
12X^2+X-1
 This is exactly the same procedure as the previous two Examples. So let's first check out the signs. (-1) this means that the signs will be different. 

So using these numbers we can figure out the factoring (3x-1)(4X+1)

You can also do something called factoring by grouping.

I will continue to use this specific example.

So from the step of using the chart. We will be using -3,4 as because that will give us 1. 

=12X^2-3X+4X-1

now we are going to use factoring by grouping

3X(4X-1)+1(4X-1)

=(4X-1)(3X+1)

This, comes out to the same answer but if you feel more comfortable using this method so you can see what is going on and why.

Some of these, you will need to be guessing and checking the factors that you think would work. But using this method, you will always get your answer.
Here is a version of the examples but may take a bit more time to complete or find the correct factors.

 This might be a little hard to figure out what is exactly going on in this picture due to the lack of quality. This photo is merely the same steps that we were using in example #3 but with possibly a little more complicated numbers.

If you would like to have some practice problems here are some great problems to review. Hint: There will be problems most likely similar to these following on a test. 

A huge hint that I got from these classes  and I think it plays a major role in why some of these might not seem to work is that if numbers are factors in a specific parenthesis, you can take out numbers so that each parenthesis is the same and you can factor by grouping. 

 

Just follow the step for any of the examples and remember to make a table. I think that the table helps a lot.

For the Review in the Book Look Through Page 176-178
For some more information on factoring quadratics and why Click HERE
For some more information for factoring quadratic equations Click HERE
And for my personal favorite site that has great information and helps a lot for reviewing, check this out and Click HERE!!!!

I hope that this summarized the first two classes of this week

Thanks guys, for reading and I hope you liked it :) 

Remember to comment!!!

Have a good weekend (hopefully with snow)