Algebra, part II factoring, quadratic equations,
exponent laws
This series of lessons is designed
to help you learn, or review, the fundamentals of algebra. In this
lesson we move on to factoring and simplifying expressions, solving
quads and dealing with exponents.
Algebra isn't as scary as some people tend to think. Up
to know we've dealt with superbasic algebra. What's coming up next
is a bit more challenging, but like all math, practice will make
this as easy as .
Let's begin by simplifying and factoring
expressions:
Take a look at: . Are you
freaking out yet? Ok, we won't deal with that one, but in short,
this sort of thing isn't such a big monster. There are ways, nice
and easy ways, of making this sort of beast become a cute little
poodle. Metaphorically speaking.
Simplification means
just that simplifying huge expressions into nicer ones. Note that
this isn't solving equations (clearly, since there's no = sign, not
even a < or > sign), so you can't just divide everything by
something or subtract something else to make the thing you're
looking at look nicer. What can we do?
One of the basic things we can do is to collect like terms. Illustrating this using a simple example: . As you can
see, that ugly thing is common to all 3 parts of the expression, so
we can make the whole thing become one by adding or subtracting the
proper coefficients. 


The more important thing though
is factoring.
Like my math teacher used to say, if you're stuck
on an ugly problem and feel like saying the F word, add a "tor" to
the end of it and you get "factor". (If you don't get this joke,
ask me later). Another simple illustration with an example:
. Yeah, since the is common, we pulled it out,
and got a much nicer thing in exchange. That's the basis of
factoring find common elements and pull them out.
An immediate and important thing to
do is learn how to factor quadratics. Quadratic expressions are
expressions with 1 variable, where the variable is raised to the
power of 2. is a good example. Note that: . Not so
scary now, is it? Practicing will make you expert at factoring this
and other expressions.
Before we go on, a few nice tricks:
Some expressions require you
to expand,
the opposite of factor. There are a few simple
expansion tricks worth remembering:
1.
2.
3.
These should help get you through the
day.
And now, let's move on to
quadratic equations
Quadratic equations
always look like this: ,
for any numbers A, B and C. Generally though, you can simplify any
expression with an to this form. Once in this form, there are 2 ways to solve
this type of equation: 

1.
Factoring: Remember this?
We can sometimes turn a nasty
into a nice situation. Once there, it's clear that either
or . These you can solve
easily. Note that you'll get 2 possible solutions that's ok,
that's what should happen most often.
2. The quadratic
formula is the second way of
solving quads. It always works, no matter what, but it can give you
nasty results and it's not as fast. The formula goes like this:
. Note that the +/
thing means you have to do it twice, once with a + and once with a
 . This will again give you 2 answers. We'll come back to this
formula later on in life to understand complex numbers.
This could be worse, right? Say,
. Well, actually, this expression
is equivalent to , but to get there you need to look at exponents and their
laws.
%{fontfamily:verdana;color:BLUE;
fontsize:16px}*Exponent laws even exponents can be made
simple%*
Yes, believe it or not, it's true.
Here is a short, exhaustive list of the rules you can use to
simplify exponents:
bq.
a. bq.
b. bq.
c. bq.
d. bq.
e.
Now you can quickly see how
.
So, with everything we learned in
this lesson, we can clean up some large, messy expressions into
nice, simple ones, and then solve them if they're in equations.
See, math isn't such an awful thing after all.
Next time, we'll get into algebra that has to do with number
theory, with some stuff about primes, complex numbers and
more. 

Thanks for reading this Welcome to
Algebra Lesson!
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