Quadratics Intro and Factorization.

A quadratic is an expression in the form . When solving a quadratic equation, we're basically trying to find values of x such that
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Factoring
Factoring is taking an expression and writing it in terms of the product of two or more expressions, or factors.

Factoring can be really simple, but also really complicated. For example, it's obvious that by factoring out the 9 in each term, and , because we "factored out" the x. THen again, it's not so obvious that (this is a famous factorization called the Sophie Germain Identity).

All the factoring this post will focus on is one called quadratic factorization (phew). It involves, well, quadratics.

Factoring is, by far, the simplest way to solve quadratics, and a very useful tool. It all starts in expansion... (Please do not use FOIL in the following problems, please distribute step by step. If you don't know FOIL, swell. :) )

Pre-problems
Expand out the following:
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What's the pattern here? (Wait until you get it before reading on).

So, every expression in the form of (x and y can be anything) is a quadratic in terms of x and y and takes the form of (note that this might not nessescarily be a quadratic in terms of other variables, say, if x=p^2, this would not be a quadratic in terms of p, but still a quadratic in terms of x and y).

To avoid confusion, only x and y will be used as variables from here on.

Expand the following:
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Notice that each expansion is a quadratic. This will always be the case for such expressions (try to see why).

Carefully look at each step you take expanding, and come up with a general expansion for , where a and b are constants (numbers).

So, you should have gotten this.

This, then, is the basis of factoring. Basically, when you factor, you look for numbers a and b that satisfy .

Here are some examples.
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We're looking for numbers a and b such that a+b=5 and ab=4. Simple guess and checkwork shows that a=1 and b=4 (or a=4 and b=1, it doesnt make a difference in this case).
Therefore, we know that . Expand it out and see if it works ;).

Now, let's try .
We're looking for numbers a and b such that a+b=-2 and ab=1. If ab is positive, either a and b are both positive, or they're both negative. But, a and b cannot have a negative sum if they're both positive, so a and b both must be negative. Guessing a=-1 and b=-1 works. Therefore, .

But wait, there might be a better way to do this! The answer gives it away, but noticing that this equation was in the form of by setting y to -1, would tell you that this expression is , right off the bat. Saved a whole bunch of effort and thinking we don't all like to do :P.
So...
REMEMBER YOUR FORMULAS AND FACTORIZATIONS.

There are also some clever "special factorizations", that are so useful that it's good to remember them all so you can recognize when to use them.
You already know the square of a binomial (two-piece expression) factorization, so the difference of squares factorization is the only new thing. Try factor . The factorization is this.

That's just it for factorization. It's not only limited to quadratics, or one variable, which can make things très compliqué, really. But as far as quadratic factorization goes, that's just it. You now have your single most important tool in solving quadratics. Congratulations.

Factorization is useful because not only is it useful, it also helps you notice patterns in math.

Now I still haven't answered the question, how do you solve quadratics with factorization? Let's say I factored a quadratic equation into . Well, it's actually really simple. All you do is take the a and b (-3 and 55), and swaps the signs around on each. So, your two (there's a max of exactly two solutions per quadratic, some have 1 and 0) solutions are 3 and -55. This works, because setting x=3 or x=-55 in the above quadratic makes one of the "factors" 0. And anything times 0 is 0. Any time you have a factored quadratic (in the form of ), just take the a and b and swap their signs (because you are looking for the value of x that, when added to a or b, makes 0).

Warning: NOT ALL QUADRATICS MAY BE FACTORED IN THIS WAY!!! Some quadratics just can't be factored- the ones that don't have any roots (solutions) to "factor into". Don't worry about those though, you will probably NEVER have to actually solve one (maybe manipulate one). Also, if the quadratic has big numbers in it, try not to try too hard :). There are other ways.

So, in summary and conclusion, factoring a quadratic is one way to solve a quadratic. It is usually done by examining what the coefficients of the quadratic have to be in terms of constants a and b, and solving for the constants, and using them to make a factorization. It this sense, quadratic factorization is "reverse distributing" or "reverse expansion".

Problems:.
Factor the following:
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For what value of x makes this NOT TRUE?

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