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 super-basic 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.
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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: |
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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.
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And now, let's move on to quadratic
equations Quadratic equations always look like this: |
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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.
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:
a.
b.
c.
d.
e.
![x ^ {\frac{1}{a}}=\sqrt[a]{x}](http://texhub.com/b/IHggXiB7XGZyYWN7MX17YX19PVxzcXJ0W2Fde3h9IA== "x ^ {\frac{1}{a}}=\sqrt[a]{x}")
Now you can quickly see how
.
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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. |
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Thanks for reading this Welcome to Algebra Lesson!
Click Here for Algebra-part-ii
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Post Comments
oLahav said – Wed, 11 Jun 2008 13:35:23 -0000 ( Flag Edit Link )
The sign ^ can be a confusing one if you’ve never seen it before. It represents exponents, so for example 2 ^ 2 is 2 to the power of 2, which is 4.
You should watch out for the brackets here: 2 ^ 2 + 1 is 4 + 1=5, but 2 ^ (2+1) is 2 ^ 3=8.
I hope this clears things up, if not start a discussion about mathematical signs in computer language, I’m sure it’ll help lots of people.